1. Find the solution of the initial-value problem: x^ay'+2x^a-1y=x^b,y(1)=1 with xgt 0 where a,b are real numbers.

Question

Answer 1

###

if

, and

if

.

Answer 2

## Step 1: Rewrite the equation### Divide the equation by

to get it in standard form:

.## Step 2: Find the integrating factor### The integrating factor is

.## Step 3: Multiply by the integrating factor### Multiply the equation by

:

.## Step 4: Integrate both sides### Notice the left side is the derivative of

. Integrating both sides with respect to

gives

, which simplifies to

assuming

.## Step 5: Solve for y### Divide by

to solve for

:

.## Step 6: Apply the initial condition### Substitute

:

, so

. Thus,

.## Step 7: Final Solution### Substitute the value of C back into the equation for y:

for

.### If

, then

. So

. Since

,

. Using

,

, so

. Thus

.

Question