1st order linear differential eq. using integrating factor

[muddyjch]

Homework Statement

Solve the inital value problem for y(x); xy′ + 7y = 2x^3 with the initial condition: y(1) = 18.
y(x) = ?

Homework Equations

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

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.

 

[mg0stisha]

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

 

[Mark44]

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

 

[cronxeh]

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.

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)