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

Given that the function $f (x) = x$ is a solution to $y''' - x^{2} y' + xy = 0$ , show that the substitution $y (x) = v (x) f (x) = v (x) x$ reduces this equation to, $xw'' + 3 w' - x^{3} w = 0$ where $w = v'$ .

Thus, it is proved that the given equation can be reduced to $xw'' + 3 w' - x^{3} w = 0 .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use the given functions to reduce the given equation to xw''+3w'-x3w=0

Given that $f (x) = x$ is a solution to $y''' - x^{2} y' + xy = 0 ...... (1)$

And

$y (x) = v (x) f (x) = v (x) x$

Now find the derivative of y for equation (1),

$\begin{matrix} y = vx \\ y' = v + xv' \end{matrix}$

Use the value $w = v'$ in the above expression,

$\begin{matrix} y' = v + xw \\ y'' = v' + xw' + w \\ y'' = w + xw' + w \\ y'' = 2 w + xw' \\ y''' = 2 w' + w' + xw'' \\ y''' = 3 w' + xw'' \end{matrix}$

Step 2: Conclusion

Substitute the all values in the equation (1),

$\begin{matrix} y''' - x^{2} y' + xy = 0 \\ 3 w' + xw'' - x^{2} (v + xw) + x (vx) = 0 \\ 3 w' + xw'' - x^{2} v - x^{3} w + x^{2} v = 0 \\ 3 w' + xw'' - x^{3} w = 0 \\ xw'' + 3 w' - x^{3} w = 0 \end{matrix}$

Thus, it is proved that the given equation can be reduced to $xw'' + 3 w' - x^{3} w = 0 .$

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Given that the function $f (x) = x$ is a solution to $y''' - x^{2} y' + xy = 0$ , show that the substitution $y (x) = v (x) f (x) = v (x) x$ reduces this equation to, $xw'' + 3 w' - x^{3} w = 0$ where $w = v'$ .

Step by Step Solution

Step 1: Use the given functions to reduce the given equation to xw''+3w'-x3w=0

Step 2: 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

Given that the function f(x)=x is a solution to y'''-x2y'+xy=0, show that the substitution y(x)=v(x)f(x)=v(x)x reduces this equation to, xw''+3w'-x3w=0 where w=v'.

Step by Step Solution

Step 1: Use the given functions to reduce the given equation to xw''+3w'-x3w=0

Step 2: 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

Given that the function $f (x) = x$ is a solution to $y''' - x^{2} y' + xy = 0$ , show that the substitution $y (x) = v (x) f (x) = v (x) x$ reduces this equation to, $xw'' + 3 w' - x^{3} w = 0$ where $w = v'$ .