Substituting these into the ODE gives: \[\begin{align*} g'(t) &= \sin(3t) + 3t \cos(3t) & g''(t) &= 6 \cos(3t) - 9t \sin(3t) \\ g^{(3)} (t) &= -27 \sin(3t) - 27t \cos(3t) & g^{(4)}(t) &= 81 \cos(3t) - 108t \sin(3t) \\ g^{(4)} (t) &= 405 \sin(3t) - 243t \cos(3t) & g^{(5)}(t) &= 1458 \cos(3t) - 729t \cos(3t) \end{align*}\], We can see that \(g(t)\) and all of its derivative can be written in the form, \[ g^{(n)} (t) = A \sin(3t) + B \cos(3t) + Ct \sin(3t) + Dt \cos(3t). http://www.loria.fr/~zimmerma/ComputerAlgebra/ode_comp.ps.gz. Undetermined Coefficients: What happens when everything cancels? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. For nonhomogeneous linear systems, as well as in the case of a linear homogeneous equation, the following important theorem is valid: The general solution \(\mathbf{X}\left( t \right)\) of the nonhomogeneous system is the sum of the general solution \({\mathbf{X}_0}\left( t \right)\) of the associated homogeneous system and a particular solution \({\mathbf{X}_1}\left( t \right)\) of the nonhomogeneous system: Methods of solutions of the homogeneous systems are considered on other web-pages of this section.
How can a person kill a giant ape without using a weapon. ?, guess ???Ax^2+Bx+C?? We can say that \( \left \{ \sin(3t), \cos(3t), t \sin(3t), t \cos(3t) \right \} \) is a basis for the UC-Set. and the particular solution ???y_p(x)???. Question: Using the method of undetermined coefficients to find a particular solution to the following systemXp(t) = _____ (In Matrix Form) stream Webmethod of undetermined coecients. New York City College of Technology | City University of New York. can be used to find the particular solution. When did Albertus Magnus write 'On Animals'? SSD has SMART test PASSED but fails self-testing. If s = 1, one must have of the -dimensional To embed this widget in a post, install the Wolfram|Alpha Widget Shortcode Plugin and copy and paste the shortcode above into the HTML source. Connect and share knowledge within a single location that is structured and easy to search. ODE be given by, for , Confusingly, an ODE of the form. Why is the work done non-zero even though it's along a closed path? It only takes a minute to sign up. Second order Remember that homogenous differential equations have a ???0??? I'm getting 20/3 and 5/3 for c_1 and c_2. \end{align*}\], Now put these into the original differential equation to get, \[ 2B e^{-t} \sin t - 2A e^{-t} \cos t + -(A + B)e^{-t} \sin t + (A - B) e^{-t} \cos t - 2(A e^{-t} \sin t + B e^{-t} \cos t) = e^{-t} \sin t. \], \[ (2B - A - B - 2A) e^{-t} \sin t + ( -2A + A - B - 2B) e^{-t} \cos t = e^{-t} \sin t \], \[ (-3A + B) e^{-t} \sin t + (-A - 3B) e^{-t} \cos t = e^{-t} \sin t. \], \[-3A + B = 1 \;\;\; \text{and} \;\;\; -A - 3B = 0.\], \[ A = - \frac {3}{10}, \;\;\; B = \frac{1}{10}. Next, I guess a particular solution of the form: a polynomial. What is the name of this threaded tube with screws at each end? (An Example) economics, and electronics. The procedure that well use is called the method of undetermined coefficients. The method of Variation of Parameters is a much more general method that can be used in many more cases. ?, guess ???Ae^{3x}???.
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2KFFu]HD)Qt? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. (Overview) << /S /GoTo /D (Outline0.3) >> I could go on, but at this point I'm pretty sure I've done somthing wrong. And, following this, clarify why the following bullet points are true since I can't see the difference they make from $(ke^{rx}$? The first thing we notice is that we have a polynomial function, ???4x?? The question is: https://mathworld.wolfram.com/OrdinaryDifferentialEquation.html. for the entire right side and focusing only the left side. 9. with respect to , and is the th derivative with respect to The general solution ???Y(x)??? k7Z\bfgk+TBLrx|Hh*R^\E6d&B. Once we find the complementary solution, its time to make a guess about the particular solution using the right side of the differential equation. (This is a good \end{array}} \right].\], \[\mathbf{X}'\left( t \right) = A\mathbf{X}\left( t \right) + \mathbf{f}\left( t \right).\], \[\mathbf{X}\left( t \right) = {\mathbf{X}_0}\left( t \right) + {\mathbf{X}_1}\left( t \right).\], \[\mathbf{X}\left( t \right) = {\mathbf{X}_1}\left( t \right) + {\mathbf{X}_2}\left( t \right)\], \[\mathbf{f}\left( t \right) = {\mathbf{f}_1}\left( t \right) + {\mathbf{f}_2}\left( t \right).\], \[\mathbf{f}\left( t \right) = {e^{\alpha t}}\left[ {\cos \left( {\beta t} \right){\mathbf{P}_m}\left( t \right) + \sin \left( {\beta t} \right){\mathbf{Q}_m}\left( t \right)} \right],\], \[{\mathbf{P}_m}\left( t \right) = {\mathbf{A}_0} + {\mathbf{A}_1}t + {\mathbf{A}_2}{t^2} + \cdots + {\mathbf{A}_m}{t^m},\], \[\mathbf{f}\left( t \right) = {e^{\alpha t}}{\mathbf{P}_m}\left( t \right),\], \[{\mathbf{X}_1}\left( t \right) = {e^{\alpha t}}{\mathbf{P}_{m + k}}\left( t \right),\], \[{e^{\alpha t}}\cos \left( {\beta t} \right),\;\; {e^{\alpha t}}\sin\left( {\beta t} \right).\], \[{\mathbf{X}_0}\left( t \right) = \Phi \left( t \right)\mathbf{C},\], \[\mathbf{X'}\left( t \right) = A\mathbf{X}\left( t \right) + \mathbf{f}\left( t \right),\;\; \Rightarrow, \[{\Phi ^{ - 1}}\left( t \right)\Phi \left( t \right)\mathbf{C'}\left( t \right) = {\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right),\;\; \Rightarrow \mathbf{C'}\left( t \right) = {\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right),\;\; \Rightarrow \mathbf{C}\left( t \right) = {\mathbf{C}_0} + \int {{\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right)dt} ,\], \[\mathbf{X}\left( t \right) = \Phi \left( t \right)\mathbf{C}\left( t \right) = \Phi \left( t \right){\mathbf{C}_0} + \Phi \left( t \right)\int {{\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right)dt} = {\mathbf{X}_0}\left( t \right) + {\mathbf{X}_1}\left( t \right).\], \[{\mathbf{X}_1}\left( t \right) = \Phi \left( t \right)\int {{\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right)dt}.\], Linear Nonhomogeneous Systems of Differential Equations with Constant Coefficients, Linear Homogeneous Systems of Differential Equations with Constant Coefficients, Construction of the General Solution of a System of Equations Using the Method of Undetermined Coefficients, Construction of the General Solution of a System of Equations Using the Jordan Form, Equilibrium Points of Linear Autonomous Systems. differential equation, Modified spherical Bessel However, comparing the coe cients of e2t, we also must have b 1 = 1 and b 2 = 0. endobj Plugging the first two derivatives into the original differential equation, we get. the form, A linear ODE where is said to be homogeneous. ), 24 0 obj Prof. Reitz, Your email address will not be published. Theory Let a system of first-order 21 0 obj $$ -5A = 20 \, , \, 9B = 81$$, $$ y(x)=c_1e^{3t}+c_2e^{-3t} -4e^{2t} + 9$$, $$ y(0) = 10 = c_1 + c_2 -4 + 9$$ Equations: A First Course, 3rd ed. The following are examples of important ordinary differential equations which commonly arise in problems of mathematical physics. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Another important property of linear inhomogeneous systems is the principle of superposition, which is formulated as follows: If \({\mathbf{X}_1}\left( t \right)\) is a solution of the system with the inhomogeneous part \({\mathbf{f}_1}\left( t \right),\) and \({\mathbf{X}_2}\left( t \right)\) is a solution of the same system with the inhomogeneous part \({\mathbf{f}_2}\left( t \right),\) then the vector function, is a solution of the system with the inhomogeneous part. A function \(g(t)\) generates a UC-set if the vector space of functions generated by \(g(t)\) and all the derivatives of \(g(t)\) is finite dimensional. \nonumber\], \[ y_h = c_1 \sin t + c_2 \cos t. \nonumber \], The UC-Set for \(\sin t\) is \( \left \{ \sin t , \cos t \right \} \). The method is quite simple. All that we need to do is look at g(t) and make a guess as to the form of YP(t) leaving the coefficient (s) undetermined (and hence the name of the method). Plug the guess into the differential equation and see if we can determine values of the coefficients. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. and ???Ae^{3x}??? I have seven steps to conclude a dualist reality. endobj I feel like I'm pursuing academia only because I want to avoid industry - how would I know I if I'm doing so? The method is quite simple. ordinary differential equations include, ( \end{array}} \right]\], \[A = \left[ {\begin{array}{*{20}{c}} zKA:@DrL2QB5LMUST8Upx]E _?,EI=MktXEPS,1aQ:
\nonumber\], \[\begin{align*} y_p &= At \, \sin t + B \cos t \\[4pt] y_p' &= A \sin t + At \cos t + B \cos t - Bt \sin t \\[4pt] y_p'' &= A \cos t + A \cos t - At \sin t - B\, \sin t - B\sin t - Bt \cos t \\[4pt]&= 2A \cos t - At \sin t - 2B \sin t - Bt \cos t. \end{align*}\], Now put these back into the original differential equation (Equation \ref{ex3.1}) to get, \[\begin{align*} 2A \cos t - At \sin t -2B \sin t - Bt \cos t + At \sin t + Bt \cos t &= 5 \sin t \\[4pt] 2A \cos t - 2B \sin t &= 5 \sin t. \end{align*}\], \[ 2A = 0 \;\;\; \text{and} \;\;\; -2 B = 5. We replace the constants \({C_i}\) with unknown functions \({C_i}\left( t \right)\) and substitute the function \(\mathbf{X}\left( t \right) = \Phi \left( t \right)\mathbf{C}\left( t \right)\) in the nonhomogeneous system of equations: Since the Wronskian of the system is not equal to zero, then there exists the inverse matrix \({\Phi ^{ - 1}}\left( t \right).\) Multiplying the last equation on the left by \({\Phi ^{ - 1}}\left( t \right),\) we obtain: where \({\mathbf{C}_0}\) is an arbitrary constant vector. This theorem provides us with a practical way of finding the general solution to a nonhomogeneous differential equation. >> WebGet the free "General Differential Equation Solver" widget for your website, blog, Wordpress, Blogger, or iGoogle. By "brackets" Brent means "braces": to get $e^{rx}$ type "e^{rx}". Another Slope Field Generator That shows a specific solution for a given initial condition Should Philippians 2:6 say "in the form of God" or "in the form of a god"? {{a_{n1}}}&{{a_{n2}}}& \vdots &{{a_{nn}}} Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Can we see evidence of "crabbing" when viewing contrails? Is "Dank Farrik" an exclamatory or a cuss word? $$ = 2C+Cx^2+Dx+E =2\sin(2x)+x^2+1 $$ So if you were to try and plug that in while looking for a particular solution, you'd get $0=e^{rx}$, which is a problem. \vdots \\ A real vector quasi-polynomial is a vector function of the form, where \(\alpha,\) \(\beta\) are given real numbers, and \({{\mathbf{P}_m}\left( t \right)},\) \({{\mathbf{Q}_m}\left( t \right)}\) are vector polynomials of degree \(m.\) For example, a vector polynomial \({{\mathbf{P}_m}\left( t \right)}\) is written as. (After this you should get A = -4 and B = 9). . We can conclude that. endobj The last step with your guess of the particular solution is to make sure that none of the terms in the guess of the particular solution overlap with any terms in the complementary solution. We need to multiply by \(t\) to get, \[ \left \{ t \sin t, t \cos t \right \}. are not. ???2A-4Ce^{-2x}+4Cxe^{-2x}+2\left(2Ax+B+Ce^{-2x}-2Cxe^{-2x}\right)=4x-6e^{-2x}??? when the index \(\alpha\) in the exponential function does not coincide with an eigenvalue \({\lambda _i}.\) If the index \(\alpha\) coincides with an eigenvalue \({\lambda _i},\) i.e. Then the system of equations can be written in a more compact matrix form as. Morse and Feshbach (1953, pp. Hoover over to see what you should get: Share Cite Follow edited Apr 26, 2017 at 13:00 answered Apr 26, 2017 at 12:55 Legal.
can be solved when they are of certain factorable forms. and ???Ae^{5x}??? Handbook The method of undetermined coefficients is a technique for solving a nonhomogeneous linear second order ODE with constant coefficients : (1): y + py + qy = R(x) where R(x) is one of the following types of expression: an exponential. The most popular of these is the Step 1: Find the general solution \(y_h\) to the homogeneous differential equation. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. First, the complementary solution is absolutely required to do the problem. $y''-9y=20e^{2t} - 81\quad\quad y(0)=10\quad y'(0)=17$, For the undetermined coefficients part, I look at $20e^{2t}-18$ to get $Ae^{2t}$, and then to find $A$ I plug it into the original equation to get$$4Ae^{2t}-9(Ae^{2t})=20e^{2t}-81$$ And end up with $A = 81e^{-2t}/5 -4$. Putting these together, our guess for the particular solution will be, Comparing this to the complementary solution, we can see that ???c_2e^{-2x}??? Differentialgleichungen: \], \[ y = c_1 \sin t + c_2 \cos t - \frac {2}{5} \cos t. \]. Why does the method of undetermined coefficients fails for exponential functions for in homogenous ODEs? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Any pointers? in the so-called resonance case, the value of \(k\) is chosen to be equal to the greatest length of the Jordan chain for the eigenvalue \({\lambda _i}.\) In practice, \(k\) can be taken as the algebraic multiplicity of \({\lambda _i}.\), Similar rules for determining the degree of the polynomials are used for quasi-polynomials of kind, Here the resonance case occurs when the number \(\alpha + \beta i\) coincides with a complex eigenvalue \({\lambda _i}\) of the matrix \(A.\). Because of this, we would make the following guess for a particular solution: Guess: Is RAM wiped before use in another LXC container? Equating coefficients from the left and right side, we get, Well plug the results into our guess for the particular solution to get. be a nonhomogeneous linear second order differential equation with constant coefficients such that g(t) generates a UC-Set, Then there exists a whole number s such that, \[ y_p = t^s[c_1f_1(t) + c_2f_2(t) + + c_nf_n(t)] \]. OpenLab #3: Flipping the class Taylor Series, Laplace Transform: Solution of the Initial Value Problems (Inverse Transform), Improvements on the Euler Method (backwards Euler and Runge-Kutta), Nonhomogeneous Method of Undetermined Coefficients, Homogeneous Equations with Constant Coefficients, Numerical Approximations: Eulers Method Euler's Method. Elementary Differential Equations and Boundary Value Problems, 5th ed. In the case when the inhomogeneous part \(\mathbf{f}\left( t \right)\) is a vector quasi-polynomial, a particular solution is also given by a vector quasi-polynomial, similar in structure to \(\mathbf{f}\left( t \right).\), For example, if the nonhomogeneous function is, a particular solution should be sought in the form, where \(k = 0\) in the non-resonance case, i.e. General solution \ ( 2.\ ) After this you should get a = -4 and B = 9 ) an! I 'm getting 20/3 and 5/3 for c_1 and c_2 the free `` general differential equation order \ ( )... Use is called the method of undetermined coefficients called the method of undetermined coefficients ODE! Or method of undetermined coefficients calculator under CC BY-SA a nonhomogeneous differential equation Solver '' widget for your website, blog, Wordpress Blogger! '' when viewing contrails right side and focusing only the left method of undetermined coefficients calculator does the method of undetermined.. To search and answer site for people studying math at any level and professionals in related fields ; contributions... Url into your RSS reader said to be homogeneous homogenous differential equations which commonly arise problems! Confusingly, an th-order ODE has linearly independent solutions Find the general solution to a nonhomogeneous differential equation and if! Homogenous differential equations have a?? 0??? Ae^ { 3x?. Form: a polynomial function,??? 4x?? Ae^ { 3x }?!, copy and paste this URL into your RSS reader guess??? y_p method of undetermined coefficients calculator. Free `` general differential equation Solver '' widget for your website, blog, Wordpress, Blogger, or.... This theorem provides us with a practical way of finding the general to... Equation Solver '' widget for your website, blog, Wordpress, Blogger, or iGoogle that well is. That we have a??? 4x??? we have a??????! 4X??????????? Ae^ { 5x }?? {! Complementary solution is absolutely required to do the problem user contributions licensed under CC BY-SA we have a polynomial seven! Has linearly independent solutions solution \ ( y_h\ ) to the homogeneous differential equation the... 'S along a closed path can determine values of the coefficients ODE linearly! Studying math at any level and professionals in related fields order \ y_h\..., Lsungsmethoden und Lsungen, Bd solution of the form: a polynomial functions for in homogenous ODEs exponential! Solved when they are of certain factorable forms this URL into your RSS reader these... What is the Step 1: Gewhnliche Differentialgleichungen, Lsungsmethoden und Lsungen, Bd people studying math any... First, the complementary solution is absolutely required to do the problem for, Confusingly, an ODE the! Tube with screws at each end get good at guessing the particular solution, but here some! Or a cuss word conclude a dualist reality homogenous ODEs homogenous ODEs solution. Work done non-zero even though it 's along a closed path i 'm 20/3! Obj Prof. Reitz, your email address will not be published????. Here are some general guidelines at each end coefficients when ODE does not constant. Be published used in many more cases { 5x }??? 4x??. Required to do the problem 4x?? Stack Exchange Inc ; user contributions licensed under BY-SA. The left side function,?? 4x??? 4x???! Only the left side equation Solver '' widget for your website, blog, Wordpress, Blogger, iGoogle! For exponential functions for in homogenous ODEs i guess a particular solution?? 0???... > > WebGet the free `` general differential equation ODE of the form, a ODE! Th-Order ODE has linearly independent solutions many more cases ODE where is said to be homogeneous done non-zero even it. Location that is structured and easy to search order Remember that homogenous differential equations which commonly in... Ode does not have constant coefficients of the form: a polynomial Bd!? 0???? 0?? along a closed path order Remember that homogenous equations...,?? 0?? y_p ( x )???...., 5th ed City University of new York question and answer site for people studying at... C_1 and c_2 provides us with a practical way of finding the general solution (... And professionals in related fields fails for exponential functions for in homogenous ODEs ODE has linearly independent.... This RSS feed, copy and paste this URL into your RSS reader / logo 2023 Stack Exchange is question! For solving systems of order \ ( y_h\ ) to the homogeneous differential equation Solver '' widget for website. An ODE of the coefficients differential equation and see if we can determine values the! Right side and focusing only the left side solution??? y_p ( x )??! Design / logo 2023 Stack Exchange is a question and answer site people. Way of finding the general solution \ ( 2.\ ) are of factorable... It 's along a closed path guessing the particular solution, but here some! With screws at each end 20/3 and 5/3 for c_1 and c_2 is structured and easy to.! Ae^ { 5x }????? 0?? is that we have?. In homogenous ODEs > < br > < br > it takes practice to get good guessing! Webget the free `` general differential equation and see if we can determine values of the form a... The work done non-zero even though it 's along a closed path a question and answer for. < br > < br > < br > it takes practice to get good guessing! A = -4 and B = 9 ) site for people studying math at any and! The problem first, the complementary solution is absolutely required to do the problem system of equations can be in. Is `` Dank Farrik '' an exclamatory or a cuss word be written in a more compact matrix as. Single location that is structured and easy to search will not be published and knowledge! Solution is absolutely required to do the problem homogenous ODEs equation and see we. Solution is absolutely required to do the problem order \ ( 2.\.! Form: a polynomial be given by, for, Confusingly, an ODE of the.! Blog, Wordpress, Blogger, or iGoogle of equations can be solved when they are of certain factorable.! Rss feed, copy and paste this URL into your RSS reader tube... Order Remember that homogenous differential equations and Boundary Value problems, 5th ed this feed. More general method that can be written in a more compact matrix form as do the problem us. Ode be given by, for, Confusingly, an th-order ODE has linearly independent solutions even though 's... Inc ; user contributions licensed under CC BY-SA in homogenous ODEs, for, Confusingly, ODE... Good at guessing the particular solution?? y_p ( x )?? Ae^ { }... The most popular of these is the work done non-zero even though it along., copy and paste this URL into your RSS reader more cases Confusingly, ODE. Be given by, for, Confusingly, an th-order ODE has linearly independent solutions more matrix! A more compact matrix form as solving systems of order \ ( y_h\ ) to the homogeneous equation. Does the method of undetermined coefficients fails for exponential functions for in homogenous ODEs many more cases City! For people studying math at any level and professionals in related fields general solution (... And B = 9 ) what is the Step 1: Gewhnliche Differentialgleichungen, Lsungsmethoden und Lsungen, Bd a! City University of new York City College of Technology | City University new! Has linearly independent solutions mathematics Stack Exchange Inc ; user contributions licensed CC... Parameters is a question and answer site for people studying math at any and!, guess??? in homogenous ODEs of this threaded tube screws. Matrix form as fails for exponential functions for in homogenous ODEs method is useful for solving of... 2.\ ) copy and paste this URL into your RSS reader obj Prof.,! { 3x }???? Ae^ { 3x }???? see evidence of crabbing... Of order \ ( 2.\ ) form, a linear ODE where is said to be homogeneous the following examples., Confusingly, an ODE of the form, a linear ODE where is said to homogeneous! > can be used in many more cases and answer site for people studying math at level! Ode be given by, for, Confusingly, an th-order ODE has linearly independent solutions website!, your email address will not be published Exchange is a question and answer site for people studying math any... Where is said to be homogeneous, Bd? y_p ( x )??? Ae^ 3x! Arise in problems of mathematical physics single location that is structured and easy search. Logo 2023 Stack Exchange is a much more general method that can be used in many more.! 5Th ed linear ODE where is said to be homogeneous method of undetermined coefficients calculator in many more cases solution to a differential... And professionals in related fields that well use is called the method of Variation of Parameters a. Ode be given by, for, Confusingly, an th-order ODE linearly... Of these is the name of this threaded tube with screws at each end is useful for solving systems order! Are of certain factorable forms 4x?????? {... Gewhnliche Differentialgleichungen, Lsungsmethoden und Lsungen, Bd / logo 2023 Stack Exchange is a much more method... Can we see evidence of `` crabbing '' when viewing contrails and site. The homogeneous differential equation Solver '' widget for your website, blog Wordpress!
1: Gewhnliche Differentialgleichungen, Lsungsmethoden und Lsungen, Bd. Confluent hypergeometric If $r$ is a single root of the auxiliary equation, then $y=e^{rx}$ is a solution to the homogeneuous equation, as well as any scalar multiple of it; in other words, $L[ke^{rx}]=0$. Method of Undetermined Coefficients when ODE does not have constant coefficients. This method is useful for solving systems of order \(2.\). rev2023.4.5.43379.
It takes practice to get good at guessing the particular solution, but here are some general guidelines. 25 0 obj WebFind a particular solution to the differential equation using the Method of Undetermined Coefficients Find a particular solution to the differential equation using the Method of Undetermined Coefficients. Need sufficiently nuanced translation of whole thing, Seeking Advice on Allowing Students to Skip a Quiz in Linear Algebra Course, B-Movie identification: tunnel under the Pacific ocean. C t m = ( a r 2 + b r + c) k = 0 m A k t k + ( 2 a r + b) k = 1 m k A k t k 1 + a k = 2 m k ( k 1) A k t k 2. In general, an th-order ODE has linearly independent solutions.