Book edition 9th

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

Pages 616 pages

ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $y'' - 6 y' + 9 y = 5 t^{6} e^{3 t}$

The particular solution is:

$y_{p} (x) = (A_{6} t^{8} + A_{5} t^{7} + A_{4} t^{6} + A_{3} t^{5} + A_{2} t^{4} + A_{1} t^{3} + A_{0} t^{2}) e^{3 t}$

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 of given differential equation.

The given differential equation is in the form of;

$ax'' + bx' + cx = e^{rt}$

To find a particular solution to the differential equation

$ay'' (x) + by' (x) + cy (x) = {Ct}^{m} e^{rt}$

Where m is a nonnegative integer, use the form;

$y_{p} (x) = t^{s} (A_{m} t^{m} + . .. + A_{1} t + A_{0}) e^{rt}$

s = 0 if r is not a root of the associated auxiliary equation;
s = 1 if r is a simple root of the associated auxiliary equation;
s = 2 if r is a double root of the associated auxiliary equation.

Step 2: Now, write the auxiliary equation of the above differential equation

The given differential equation is,

$y'' - 6 y' + 9 y = 5 t^{6} e^{3 t} . ..... (1)$

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

$y'' - 6 y' + 9 y = 0$

The auxiliary equation for the above equation,

$r^{2} - 6 r + 9 = 0$

Step 3: Now find the roots of auxiliary equation

Solve the auxiliary equation,

$\begin{matrix} r^{2} - 6 r + 9 = 0 \\ {(r - 3)}^{2} = 0 \end{matrix}$

The roots of auxiliary equation are,

$r_{1} = 3, & r_{2} = 3$

The complimentary solution of the given equation is,

$y_{c} = c_{1} e^{3 t} + c_{2} {te}^{3 t}$

Step 4: Final conclusion

To find a particular solution to the differential equation;

$ay'' (x) + by' (x) + cy (x) = {Ct}^{m} e^{rt}$

Compare with the given differential equation,

$y'' - 6 y' + 9 y = 5 t^{6} e^{3 t}$

Condition satisfied,

M=6, s = 2 if r = 3 is a double root of the associated auxiliary equation.

Therefore, the particular solution of equation,

$\begin{matrix} y_{p} (x) = t^{2} (A_{6} t^{6} + A_{5} t^{5} + A_{4} t^{4} + A_{3} t^{3} + A_{2} t^{2} + A_{1} t + A_{0}) e^{3 t} \\ y_{p} (x) = (A_{6} t^{8} + A_{5} t^{7} + A_{4} t^{6} + A_{3} t^{5} + A_{2} t^{4} + A_{1} t^{3} + A_{0} t^{2}) e^{3 t} \end{matrix}$

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $y'' - 6 y' + 9 y = 5 t^{6} e^{3 t}$

Step by Step Solution

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

Step 2: Now, write the auxiliary equation of the above differential equation

Step 3: Now find the roots of auxiliary equation

Step 4: Final conclusion

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) y''-6y'+9y=5t6e3t

Step by Step Solution

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

Step 2: Now, write the auxiliary equation of the above differential equation

Step 3: Now find the roots of auxiliary equation

Step 4: Final conclusion

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $y'' - 6 y' + 9 y = 5 t^{6} e^{3 t}$