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.
$x'' (t) - 4 x' (t) + 4 x (t) = {te}^{2 t}$

The particular solution is $x_{p} = \frac{1}{6} t^{3} e^{2 t} .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Firstly, write the auxiliary equation of the above differential equation

Consider the given differential equation,

$x'' (t) - 4 x' (t) + 4 x (t) = {te}^{2 t} \dots (1)$

Write the homogeneous differential equation of the equation (1),

$x'' (t) - 4 x' (t) + 4 x (t) = 0$

The auxiliary equation for the above equation,

$m^{2} - 4 m + 4 = 0$

Step 2: Now find the roots of the auxiliary equation

Solve the auxiliary equation,

$\begin{matrix} m^{2} - 4 m + 4 = 0 \\ {(m - 2)}^{2} = 0 \end{matrix}$

The roots of the auxiliary equation are;

$m_{1} = 2, & m_{2} = 2$

The complementary solution of the given equation is;

$x_{c} = c_{1} e^{2 t} + c_{2} {te}^{2 t}$

Step 2: Find a particular solution.

Therefore, the particular solution of equation (1),

$x_{p} = t^{2} (At + B) e^{2 t} \dots (2)$

Now find the derivative of the above equation,

$\begin{matrix} x_{p}' (t) = (3 {At}^{2} + 2 Bt) e^{2 t} + 2 t^{2} (At + B) e^{2 t} \\ x_{p}' (t) = (3 {At}^{2} + 2 Bt + 2 {At}^{3} + 2 t^{2} B) e^{2 t} \\ x_{p}'' (t) = (12 {At}^{2} + 8 Bt + 6 At + 4 {At}^{3} + 4 t^{2} B + 2 B) e^{2 t} \end{matrix}$

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

$\begin{matrix} x''_{p} (t) - 4 x_{p}' (t) + 4 x_{p} (t) = {te}^{2 t} \\ (12 {At}^{2} + 8 Bt + 6 At + 4 {At}^{3} + 4 t^{2} B + 2 B) e^{2 t} - 4 (3 {At}^{2} + 2 Bt + 2 {At}^{3} + 2 t^{2} B) e^{2 t} + 4 t^{2} (At + B) e^{2 t} = {te}^{2 t} \\ {te}^{2 t} (6 A) + 2 {Be}^{2 t} = {te}^{2 t} \end{matrix}$

Step 4: Final conclusion.

Comparing all coefficients of the above equation,

$\begin{matrix} 6 A = 1 \Rightarrow A = \frac{1}{6} \\ B = 0 \end{matrix}$

Therefore, the particular solution of equation (1),

$\begin{matrix} x_{p} = t^{2} (At + B) e^{2 t} \\ x_{p} = t^{2} (\frac{}{} t + 0) e^{2 t} \\ x_{p} = \frac{}{} t^{3} e^{2 t} \end{matrix}$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Find a particular solution to the differential equation.
$x'' (t) - 4 x' (t) + 4 x (t) = {te}^{2 t}$

Step by Step Solution

Step 1: Firstly, write the auxiliary equation of the above differential equation

Step 2: Now find the roots of the auxiliary equation

Step 2: Find a particular solution.

Step 4: Final conclusion.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Find a particular solution to the differential equation.x''(t)-4x'(t)+4x(t)=te2t

Step by Step Solution

Step 1: Firstly, write the auxiliary equation of the above differential equation

Step 2: Now find the roots of the auxiliary equation

Step 2: Find a particular solution.

Step 4: Final conclusion.

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Find a particular solution to the differential equation.
$x'' (t) - 4 x' (t) + 4 x (t) = {te}^{2 t}$