# Find integrating factor and solve the equation 3

y' + y = e^x ; y(0) = 1

1st, i calculate the integrating factor...
u(x) = e^x

times the integrating factor with DE...

y'e^x + ye^x = e^2x

dy/dx e^x + ye^x = e^2x

d/dx ye^x = e^2x

ye^x = ∫ e^2x dx
.........= 1/2 e^2x + C

y = 1/2 e^x + C

the problem here, i didn't get the answer given which is..
y = 1/2 (e^x + e^-x)

## Answers and Replies

vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
You made an algebra mistake in the last step when solving for y.

ok.. here is my mistake...
ye^x = ∫ e^2x dx
.........= 1/2 e^2x + C

so.. when am i going to solve C value?

vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
ok.. here is my mistake...
ye^x = ∫ e^2x dx
.........= 1/2 e^2x + C

so.. when am i going to solve C value?

Up to there is correct. Your error was in the very last step you wrote when you solved for y.

You can solve for C any time you want. Most of the time, it's done as the final step.