Find the general solution?

Question

Find the general solution of the differential equation y'=y+x^2.

 

Answer: Ce^x-x^2-2x-2

 

Please show all your work.

Answer 1

We approach this through the use of an integrating factor.

 

1. Rearrange the terms

y' - y = x²

 

2. Find the integrating factor I

I = exp( int [0,x] -1 dt ) = exp(-x)

 

3. Using the integrating factor I and simplifying the expression

d/dx (yI) = x²I

 

4. Integrate both sides

y exp(-x) = int x² exp(-x) dx

 

5. Integrate the right hand side by parts (see note below)

y exp(-x) = -exp(-x) (x² + 2x + 2) + C

 

6. Multiply both sides by exp(x)

y = C exp(x) - x² - 2x - 2

 

 

 

Note: Integrate x² exp(-x) by parts, where int u dv = uv - int v du

1. Let u = x² and dv = exp(-x)

2. Then du = 2x and v = -exp(-x)

3. int x² exp(-x) dx = -exp(-x) x² + int exp(-x) 2x dx

3a. Now let u = 2x and dv = exp(-x)

3b. Then du = 2 and v = -exp(-x)

3c. int exp(-x) 2x dx = -exp(-x) 2x + int exp(-x) 2 dx = -exp(-x) 2x - 2 exp(-x) + C

4. int x² exp(-x) dx = -exp(-x) x² - exp(-x) 2x - exp(-x) 2 + C

5. int x² exp(-x) dx = -exp(-x) (x² + 2x + 2) + C