Chapter 8: Ordinary Differential Equations

Find the inverse transforms of the functions$F\left(p\right)$.

role="math" localid="1664277165358" $\frac{3p+2}{3p^2+5p-2}$

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection. Also find a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form.

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{10}\mathit{y}\mathbf{=}\mathbf{100}\mathit{c}\mathit{o}\mathit{s}\mathbf{4}\mathit{x}$

Question: Solve (12.12)andto get (12.15). Hint: Use Cramer's rule (Chapter 3, Section 3); note that the denominator determinant is the Wronskian [Chapter 3 , equation (8.5)) of the functionssin xand cos x.

Question: find the solution of ( 12.7 )with$\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathit{y}\mathbf{(}\mathit{\pi }\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{0}$when the forcing function is given f ( x ).

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{x}$

Question: Find the solution of ( 12.7 ) with $\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathit{y}\mathbf{(}\mathit{\pi }\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{0}$when the forcing function is given f(x).

f (x) = sec x.

Question: Find the solution of ( 12.7 )with$\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathit{y}\mathbf{(}\mathit{\pi }\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{0}$when the forcing function is given$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$.$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\{\begin{array}{cc}x,& 0<x<\pi /4\\ \pi /2-x,& \pi /4<x<\pi /2.\end{array}$

Question: (a) Given that ${\mathit{\psi }}_{\mathbf{1}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$and y₂(x)are solutions of (12.19)with f(x)=0, and that${\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{b}\mathbf{\right)}\mathbf{=}\mathbf{0}$, find the Green function [as in (12.11) to (12.16)] and so obtain the solution (12.20). Then find the particular solution (12.21) as discussed for (12.18) and (12.21).

(b) The method of variation of parameters is an elementary way of finding a particular solution of (12.19) when you know the solutions of the homogeneous equation. Show as follows that this method leads to the same result (12.21) as the Green function method. Start with the known solution of the homogeneous equation, say$\mathit{y}\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{y}}_{\mathbf{1}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{y}}_{\mathbf{2}}$and allow the "constants" to be functions ofxto be determined so thatysatisfies (12.19). (The c's are the "parameters" which are to be "varied" in the expression "variation of parameters".) You want to find y'and y"to substitute into (12.19). First find y'and set the sum of the terms involving derivatives of the c's equal to zero. Differentiate the rest of y'again to get y". Now substitute y, y'and y"into (12.19)and use the fact that y₁and y₂both satisfy the homogeneous equation [that is, (12.19) with f(x) = 0 }. You should have the two equations:

$\begin{array}{l}{\mathbf{c}}_{\mathbf{1}}^{\mathbf{\text{'}}}{\mathbf{\xi }}_{\mathbf{1}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}^{\mathbf{\text{'}}}{\mathbf{y}}_{\mathbf{2}}\mathbf{=}\mathbf{0}\\ {\mathbf{c}}_{\mathbf{1}}^{\mathbf{\text{'}}}{\mathbf{\xi }}_{\mathbf{1}}^{\mathbf{\text{'}}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{y}}_{\mathbf{2}}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\end{array}$

Solve this pair of equations for C₁and C₂'[say by determinants, and note that the denominator determinant is the Wronskian as in (12.20) and (12.21)]. Write the indefinite integrals for c₁and c₂, and write$\mathit{y}\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{y}}_{\mathbf{1}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{y}}_{\mathbf{2}}$to get (12.21).

Question: Use the given solutions of the homogeneous equation to find a particular solution of the given equation. You can do this either by the Green function formulas in the text or by the method of variation of parameters

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}\mathit{y}\mathbf{=}\mathit{s}\mathit{e}\mathit{c}\mathit{h}\mathit{x}\mathbf{;}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathit{x}\mathbf{,}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathit{x}$

Question: Use the given solutions of the homogeneous equation to find a particular solution of the given equation. You can do this either by the Green function formulas in the text or by the method of variation of parameters

${\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}\mathbf{2}\mathit{x}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{=}\mathit{x}\mathit{l}\mathit{n}\mathit{x}\mathbf{;}\mathit{x}\mathbf{,}{\mathit{x}}^{\mathbf{2}}$

Question: Use the given solutions of the homogeneous equation to find a particular solution of the given equation. You can do this either by the Green function formulas in the text or by the method of variation of parameters

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}\mathbf{2}\left(cs{c}^{2}x\right)\mathit{y}\mathbf{=}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{x}\mathbf{;}\mathit{c}\mathit{o}\mathit{t}\mathit{x}\mathbf{,}\mathbf{1}\mathbf{-}\mathit{x}\mathit{c}\mathit{o}\mathit{t}\mathit{x}$