Explanation Given: The differential equation is d 2 y d x 2 = 1 x d y d x − 4 x 2 y . Approach: A second-order homogeneous equation in the form a x 2 y ″ ( x ) + b x y ′ ( x ) + c y ( x ) = 0 is called Cauchy - Euler equation, where a ≠ 0 and b and c are real numbers. If the equation has a singular point then it has solution in the form of y = x r . If the roots of equation a r 2 + b r + c = 0 are real such that r = r 1 = r 2 , then general solution is y ( t ) = c 1 e r t + c 2 t e r t If the roots of equation a r 2 + b r + c = 0 are real such that r 1 ≠ r 2 then general solution is y ( t ) = c 1 e r 1 t + c 2 e r 2 t If the roots of equation a r 2 + b r + c = 0 are imaginary in form of r 1 , 2 = α ± β i then general solution is y ( t ) = e a t ( c 1 cos β t + c 2 sin β t ) Calculation: Consider the given problem. d 2 y d x 2 = 1 x d y d x − 4 x 2 y d 2 y d x 2 − 1 x d y d x + 4 x 2 y = 0 … ( 1 ) As x > 0 multiply equation ( 1 ) by x 2 . x 2 d 2 y d x 2 − x d y d x + 4 y = 0 ... ( 2 ) Let, w ( r , x ) = x r and L is the left hand side of the equation. Differentiate w ( r , x ) = x r upto second derivative. d 2 d r 2 { L [ w ] ( x ) } = d 2 d r 2 x r = d d r r x r − 1 = r ( r − 1 ) x r − 2 Substitute r ( r − 1 ) x r − 2 for y ″ r x r − 1 for y ′ and x r for y in equation ( 2 )

Fundamentals of Differential Equations [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

Author: Saff, Edward B., Snider, Arthur David, Nagle, R. Kent

Publisher: Addison Wesley

See similar textbooks

Videos

Question

Chapter 8.5, Problem 6E

To determine

To find:

The general solution of a differential equation by substitution of y=xr, for x>0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.5, Problem 5E

Chapter 8.5, Problem 7E

Students have asked these similar questions

1.2.7. (-) Prove that a bipartite graph has a unique bipartition (except for interchang- ing the two partite sets) if and only if it is connected.

Sx. KG A3 is collection of Countin uous function on a to Polgical Which separates Points Srem closed set then the toplogy onx is the weak toplogy induced by the map fx. Prove that using dief speParts Point If B closed and x&B in X then for some xеA fx(x) € fa(B). If (π Xx, prodect) is prodect space KEA S Prove s. BxXx (πh Bx) ≤ πTx B x Prove is an A is finte = (πT. Bx) = πT. Bå KEA XEA

AP1.2 A child is 40 inches tall, which places her at the 90th percentile of all children of similar age. The heights for children of this age form an approximately Normal distribution with a mean of 38 inches. Based on this information, what is the standard deviation of the heights of all children of this age? 0.20 inches (c) 0.65 inches (e) 1.56 inches 0.31 inches (d) 1.21 inches

Chapter 8 Solutions

Fundamentals of Differential Equations [With CDROM] - 7th Edition

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The general solution of a differential equation by substitution of y = x r , for x > 0 .

Solution Summary: The author explains the general solution of a differential equation by substitution of y=xr.

Videos

To find:

The general solution of a differential equation by substitution of y=xr, for x>0.

Want to see the full answer?

Chapter 8 Solutions

The general solution of a differential equation by substitution of y = x r , for x &gt; 0 .

Solution Summary: The author explains the general solution of a differential equation by substitution of y=xr.

Videos

To find: The general solution of a differential equation by substitution of y=xr, for x>0.

Want to see the full answer?

Chapter 8 Solutions

The general solution of a differential equation by substitution of y = x r , for x > 0 .

To find:

The general solution of a differential equation by substitution of y=xr, for x>0.