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.
$2 z'' + z = 9 e^{2 t}$

The particular solution of the given differential equation is $z_{p} (x) = e^{2 t} .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: firstly, write the auxiliary equation of the given differential equation

The given differential equation is,

$2 z'' + z = 9 e^{2 t} \dots (1)$

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

$2 z'' + z = 0$

The auxiliary equation for the above equation,

$2 m^{2} + 1 = 0$

Step 2: Now find the roots of the auxiliary equation

Solve the auxiliary equation,

$\begin{matrix} 2 m^{2} + 1 = 0 \\ m^{2} = \frac{- 1}{2} \\ m = \pm i \sqrt{\frac{1}{2}} \end{matrix}$

The roots of the auxiliary equation are,

$m_{1} = i \sqrt{\frac{1}{2}}, m_{2} = - i \sqrt{\frac{1}{2}}$

The complementary solution of the given equation is,

$Z_{c} (x) = c_{1} \cos (\sqrt{\frac{1}{2}} t) + c_{2} \sin (\sqrt{\frac{1}{2}} t)$

Step 3: Final conclusion, find a particular solution to the differential equation

According to the method of undetermined coefficients, assume, the particular solution of equation (1),

$z_{p} (x) = {Ae}^{2 t} . ..... (2)$

Now find the derivative of the above equation,

$\begin{matrix} z_{p}' (x) = 2 {Ae}^{2 t} \\ z_{p}'' (x) = 4 {Ae}^{2 t} \end{matrix}$

From the equation (1),

$\begin{matrix} 2 z_{p}'' + z_{p} = 9 e^{2 t} \\ 2 (4 {Ae}^{2 t}) + {Ae}^{2 t} = 9 e^{2 t} \\ 9 {Ae}^{2 t} = 9 e^{2 t} \\ A = 1 \end{matrix}$

Substitute the value of A in the equation (2),

$z_{p} (x) = e^{2 t}$

Therefore, the particular solution of equation (1),

$z_{p} (x) = e^{2 t}$

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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.
$2 z'' + z = 9 e^{2 t}$

Step by Step Solution

Step 1: firstly, write the auxiliary equation of the given differential equation

Step 2: Now find the roots of the auxiliary equation

Step 3: Final conclusion, find a particular solution to the differential equation

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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.2z''+z=9e2t

Step by Step Solution

Step 1: firstly, write the auxiliary equation of the given differential equation

Step 2: Now find the roots of the auxiliary equation

Step 3: Final conclusion, find a particular solution to the differential equation

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Find a particular solution to the differential equation.
$2 z'' + z = 9 e^{2 t}$