# Thread: First order differential and linear equations

1. ## First order differential and linear equations

Hello guys,

I need help with two of the most weird equations i have ever seen.

Problem 1:
Solve the initial value problem

xy' + 3y = x^7, y'(0) = 3.

Problem 2:
Solve the initial value problem

(x^2)y' + y = 1, y(1) = 2.

2. Originally Posted by tapiaghector
Hello guys,

I need help with two of the most weird equations i have ever seen.

Problem 1:
Solve the initial value problem

xy' + 3y = x^7, y'(0) = 3.

These quations aren't very weird at all. You need to use the integrating factor method for both. They just need a little re-arranging first.

$xy' + 3y = x^7$

divide both sides by x

$y' + \frac{3}{x}y = x^6$

Then follow the steps in this post

http://www.mathhelpforum.com/math-he...equations.html

3. hey thanks my friend, i have another question, how do you evaluate y'(0) = 3?

4. Originally Posted by tapiaghector
hey thanks my friend, i have another question, how do you evaluate y'(0) = 3?
After you've found your general solution you differentiate it. Then you substitute x = 0 and equate the result to 3. Then you solve for your arbitrary constant of integration.

5. Originally Posted by tapiaghector
hey thanks my friend, i have another question, how do you evaluate y'(0) = 3?

This is to be evaluated after you have a general solution.

$y=\frac{ \int x^6 e^{3ln(x)} dx}{e^{3ln(x)}}$