- #1
gulsen
- 217
- 0
I have a basic differential equation:
[tex]\frac{dy}{dx} = x + y, y(0) = 1[/tex]
Now, when I try to solve this by making it exact
[tex]\mu \frac{dy}{dx} + \mu y = \mu x[/tex]
I get [tex]\mu = e^{-x}[/tex] and solution [tex]-x-1[/tex]. This doesn't satisfy the initial condition. But when I try to solve it as a non-homogenous equation as:
[tex]\frac{dy}{dx} + y= x[/tex]
I get
[tex]y_p = 2e^x, y_c = -x-1[/tex]
so
[tex]y = 2e^x-x-1[/tex]
Which seems to be a correct & full solution. What was I missing in the first try?
[tex]\frac{dy}{dx} = x + y, y(0) = 1[/tex]
Now, when I try to solve this by making it exact
[tex]\mu \frac{dy}{dx} + \mu y = \mu x[/tex]
I get [tex]\mu = e^{-x}[/tex] and solution [tex]-x-1[/tex]. This doesn't satisfy the initial condition. But when I try to solve it as a non-homogenous equation as:
[tex]\frac{dy}{dx} + y= x[/tex]
I get
[tex]y_p = 2e^x, y_c = -x-1[/tex]
so
[tex]y = 2e^x-x-1[/tex]
Which seems to be a correct & full solution. What was I missing in the first try?