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Q12E

Expert-verified

Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 180

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Find a particular solution to the differential equation.
$2 x' + x = 3 t^{2}$

The particular solution is $x_{p} (t) = 3 t^{2} - 12 t + 24 .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use the method of undetermined coefficients to find a particular solution to the differential equation.

Consider the given differential equation,

$2 x' + x = 3 t^{2} \dots (1)$

According to the method of undetermined coefficients, the particular solution of the differential equation;

$ax'' + bx' + cx = {dt}^{m}, m = 0, 1, 2, 3, . ..$

It is of the form $x_{p} (t) = A_{m} t^{m} + A_{m - 1} t^{m - 1} + . .. + A_{1} t + A_{0}$

Comparing the above equation with equation (1),

We get, m = 2

Step 2: Find a particular solution to the differential equation for m = 2

Therefore, the particular solution of equation (1),

$x_{p} (t) = A_{2} t^{2} + A_{1} t + A_{0} \dots (2)$

Now find the derivative of above equation,

$x_{p}' (t) = 2 A_{2} t + A_{1}$

From the equation (1), substitute the value of $x_{p}' (t)$ and $x_{p} (t)$ , we get

$\begin{matrix} 2 x_{p}' + x_{p} = 3 t^{2} \\ 2 [2 A_{2} t + A_{1}] + A_{2} t^{2} + A_{1} t + A_{0} = 3 t^{2} \\ 4 A_{2} t + 2 A_{1} + A_{2} t^{2} + A_{1} t + A_{0} = 3 t^{2} \\ A_{2} t^{2} + [4 A_{2} + A_{1}] t + 2 A_{1} + A_{0} = 3 t^{2} \end{matrix}$

Step 3: Final conclusion.

Comparing the all coefficients of the above equation,

$\begin{matrix} A_{2} = 3 \\ 4 A_{2} + A_{1} = 0 \dots (3) \\ 2 A_{1} + A_{0} = 0 \dots (4) \end{matrix}$

Substitute the value of $A_{2}$ in the equation (3),

$\begin{matrix} 4 (3) + A_{1} = 0 \\ A_{1} = - 12 \end{matrix}$

Substitute the value of $A_{1}$ in the equation (4),

$\begin{matrix} 2 (- 12) + A_{0} = 0 \\ A_{0} = 24 \end{matrix}$

Substitute the value of $A_{0}, A_{1}$ and $A_{2}$ in the equation (2),

$x_{p} (t) = 3 t^{2} - 12 t + 24$

Therefore, the particular solution of equation (1),

$\begin{matrix} x_{p} (t) = A_{2} t^{2} + A_{1} t + A_{0} \\ x_{p} (t) = 3 t^{2} + (- 12) t + 24 \\ x_{p} (t) = 3 t^{2} - 12 t + 24 \end{matrix}$

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