# Homework Help: 1st order linear differential eq. using integrating factor

1. Jun 3, 2010

### muddyjch

1. The problem statement, all variables and given/known data
Solve the inital value problem for y(x); xy′ + 7y = 2x^3 with the initial condition: y(1) = 18.
y(x) = ?

2. Relevant equations
dy/dx +P(x)y=Q(x), integrating factor=e^∫P(x) dx

3. The attempt at a solution
Multiplied all terms by 1/x to get it in correct form dy/dx+7y/x=2x^2
integrating factor=e^∫7/x dx=x^7
here is where i get lost do i multiply everything back into the original eq. or am i already off. I am not getting the correct answer.

2. Jun 3, 2010

### mg0stisha

Maybe it's just a typo, but you might want to revise your integrating factor.

3. Jun 3, 2010

### Staff: Mentor

Right.
$$\int \frac{7 dx}{x} \neq x^7$$

4. Jun 3, 2010

### cronxeh

Once you get the correct integrating factor M(x) you will multiply it out over entire equation, on both sides. Your equation will then reduce to y(x)M(x)=Integral(M(x)Q(x)dx) + C. to get y(x) you will integrate and plug in the initial conditions.

y(x) = (Integral(M(x)Q(x)dx) + C) / M(x)

y(x)=(x^10+89)/(5x^7)