   Chapter 6.1, Problem 41E

Chapter
Section
Textbook Problem

Question: Verify that the general solution 〈 em 〉 y = C 1 x + C 2 x 3 , 〈 / em 〉 satisfies the differential equation 〈 em 〉 x 2 y & # 8242 ; ' & # 8242 ; ' & # 8722 ; − 3 x y & # 8242 ; ' + 3 y = 0 , x > 0. 〈 / em 〉 Then find the particular solution that satisfies the initial conditions 〈 em 〉 y = 0 〈 / em 〉 when 〈 em 〉 x = 2 〈 / em 〉 and 〈 em 〉 y & # 8242 ; ' = 4 〈 / em 〉 when 〈 em 〉 x = 2. 〈 / p 〉 〈 / em 〉

(i)

To determine
Whether the general solution y=C1x+C2x3 satisfies the differential equation x2y''3xy'+3y=0;x>0

Explanation

Given:

i) The function y=C1x+C2x3.

ii) The differential equation x2y''3xy'+3y=0;x>0.

Explanation:

The general solution y=C1x+C2x3 has been provided.

On differentiating both sides of y with respect to x, the following is derived:

y'=C1+3C2x2............................. (a)

On differentiating y' with respect to x , the following is derived:

y''=0+6C2x=6C2x y''=6C2xC2=y''6x …...…...…...…...…...…...…...…...(b)

From (a) and (b) we get

y'=C1+3.(y''6x).x2=C1+xy''2C1=y'xy''2

(ii)

To determine

To calculate: The specific solution with initial conditions y=0 when x=2 and y'=4 when x=2.

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