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

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. $y'' - y' + y = {(e^{t} + t)}^{2}$

Yes.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use the method of undetermined coefficients

The given differential equation is in the form of $a x'' + b x' + c x = e^{r t}$

According to the method of undetermined coefficients,

To find a particular solution to the differential equation:

$a y'' (x) + b y' (x) + c y (x) = C t^{m} e^{r t}$

Where m is a non-negative integer, use the form

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

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,

$y'' - y' + y = {(e^{t} + t)}^{2} y'' - y' + y = e^{2 t} + t^{2} + 2 t e^{t} . . . (1)$

Write the homogeneous differential equation of equation (1),

$y'' - y' + y = 0$

The auxiliary equation for the above equation,

$r^{2} - r + 1 = 0$

Step 3: Now find the roots of the auxiliary equation

Solve the auxiliary equation,

$r^{2} - r + 1 = 0 r = \frac{1 \pm \sqrt{1 - 4}}{2} r = \frac{1 \pm i \sqrt{3}}{2}$

The roots of the auxiliary equation are,

$r_{1} = \frac{1 + i \sqrt{3}}{2}, r_{2} = \frac{1 - i \sqrt{3}}{2}$

Step 4: Final Conclusion

To find a particular solution to the differential equation:

$a y'' (x) + b y' (x) + c y (x) = C t^{m} e^{r t}$

Compare with the given differential equation,

$y'' - y' + y = e^{2 t} + t^{2} + 2 t e^{t}$

The first condition is satisfied, one has:

M=0 and r = 2 are not a root of the associated auxiliary equation;

s = 0 if r is not a root of the associated auxiliary equation;

Therefore, the particular solution of the equation,

$y_{p} (x) = t^{s} (A_{m} t^{m} + . . . + A_{1} t + A_{0}) e^{r t} y_{p} (x) = t^{0} (A_{0}) e^{2 t} y_{p} (x) = A_{0} e^{2 t}$

The second condition satisfied,

One has,

M=1 and r = 1 are not a root of the associated auxiliary equation;

s = 0 if r is not a root of the associated auxiliary equation;

Hence, the particular solution of the equation,

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

The third condition is satisfied, one has:

M=2 and r = 0 are not a root of the associated auxiliary equation;

s = 0 if r is not a root of the associated auxiliary equation;

Accordingly, the particular solution of the equation,

$y_{p} (x) = t^{s} (A_{m} t^{m} + . . . + A_{1} t + A_{0}) e^{r t} y_{p} (x) = t^{0} (A_{2} t^{2} + A_{1} t + A_{0}) e^{(0) t} y_{p} (x) = A_{2} t^{2} + A_{1} t + A_{0}$

R.H.S. of the equation $t^{2}, e^{2 t}$ and $2 t e^{t}$ is the combination of polynomials, exponentials, sines or cosines or product of these t function.

So, the method of undetermined coefficients can be applied.

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. $y'' - y' + y = {(e^{t} + t)}^{2}$

Step by Step Solution

Step 1: Use the method of undetermined coefficients

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

Step 3: Now find the roots of the 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

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. y''-y'+y=et+t2

Step by Step Solution

Step 1: Use the method of undetermined coefficients

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

Step 3: Now find the roots of the 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

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Calculus

Pure Maths

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. $y'' - y' + y = {(e^{t} + t)}^{2}$