Question

PLEASE ANSWER THESE QUESTION, ITS URGENT........

1. Solve the following ODE using the power series method:
((x^2)- 1) y" + 3xy"+ xy = 0

2. Obtain all the first and second order partial derivatives of the function:
f(x,y) = ln (x^2 + y^2)

Answer 1

sorry i cant understamd this question..

Answer 2

1.Let y(x)=∑n=0∞anxn. Then we get that

y′(x)=∑n=0∞nanxn−1

3xy′(x)=∑n=0∞3nanxn

y′′(x)=∑n=0∞n(n−1)anxn−2

xy′′(x)=∑n=0∞n(n−1)anxn−1=∑n=0∞n(n+1)an+1xn

x2y′′(x)=∑n=0∞n(n−1)anxn

The ODE is

xy′′−x2y′′−3xy′−y=0

Plugging in the appropriate series expansions, we get that

∑n=0∞(n(n+1)an+1−n(n−1)an−3nan−an)xn=0

Hence, we get that

n(n+1)an+1=(n(n−1)+3n+1)an=(n+1)2an⟹an+1=n+1nan

First note that a0=0. Choose a1 arbitrarily. Then we get that a2=2a1, a3=3a1, a4=4a1 and in general, an=na1. Hence, the solution is given by

y(x)=a1(x+2x2+3x3+⋯)

This power series is valid only within |x|<1. In this region, we can simplify the power series to get

y(x)=a1x(1+2x+3x2+⋯)=a1xddx(x+x2+x3+⋯)=a1xddx(x1−x)=a1x(1−x)2
2.The notation ∂2f∂x∂y∂2f∂x∂y is the same as

∂∂x∂f∂y=∂∂x2yx2+y2+3=2y∂∂x1x2+y2+3=2y−2x(x2+y2+3)2=−4xy(x2+y2+3)2