Find a particular solution of the following non-homogeneous equations by theselected method.(a) Undetermined coefficients: y" + 4y = 8t2 + 4t.(b) Variation of parameters: y" +9y = sec(3t).%3D

Answered: Find a particular solution of the…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Find a particular solution of the following non-homogeneous equations by the
selected method.
(a) Undetermined coefficients: y" + 4y = 8t2 + 4t.
(b) Variation of parameters: y" +9y = sec(3t).
%3D

Expert Solution

part a

Given equation is

$y'' + 4 y = 8 t^{2} + 4 t$ (i)

It is a second order linear non-homogeneous differential equation.

Now, non-homogeneous part of the given equation is the function $8 t^{2} + 4 t$ . Let us consider that the particular integral is

$y_{p} = A t^{2} + B t + C$

where A, B and C are undetermined constants to be determined. Differentiating $y_{p}$ w.r.t t,

$y_{p}' = 2 A t + B y_{p}'' = 2 A$

Substituting in eq. (i)

$\begin{array}{rcl} 2 A + 4 (A t^{2} + B t + C) & = & 8 t^{2} + 4 t \\ 4 A t^{2} + 4 B t + (2 A + 4 C) & = & 8 t^{2} + 4 t (i i) \end{array}$

Comparing co-efficients of $t^{2}$ and t from both sides of (ii),

$\begin{array}{rcl} 4 A & = & 8 \\ A & = & 2 \\ 4 B & = & 4 \\ B & = & 1 \\ 2 A + 4 C & = & 0 \\ 4 C & = & - 4 \\ C & = & - 1 \end{array}$

So, the particular integral is

$y_{p} = 2 t^{2} + t - 1$

Hence, particular integral of the given equation is $y_{p} = 2 t^{2} + t - 1$

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Find a particular solution of the following non-homogeneous equations by the selected method. (a) Undetermined coefficients: y' + 4y = 8t2 + 4t. (b) Variation of parameters: y" + 9y = sec(3t).