What is the general solution of the differential equation dy/dx- 2xy = x ?
1 Answer
y = 3/2e^(x^2 - 1) - 1/2
Explanation:
We have:
dy/dx- 2xy = x
Which we can write as:
dy/dx = 2xy+ x
:. dy/dx = (2y+ 1)x
:. 1/(2y+ 1) dy/dx = x
Which is a first order separable differential equation, so we can "separate the variables" to get:
int \ 1/(2y+ 1) \ dy = int \ x \ dx
Integrating we get, the General Solution:
1/2ln|2y+1| = 1/2x^2 + C
Applying the initial condition
1/2ln3 = 1/2 + C => C = 1/2ln3 - 1/2
So we can write an implicit particular solution as:
1/2ln|2y+1| = 1/2x^2 + 1/2ln3 - 1/2
We typically require an explicit solution, so we can rearrange as follows:
ln|2y+1| = x^2 + ln3 - 1
:. |2y+1| = e^(x^2 + ln3 - 1)
Noting that the exponential function is positive over its entire domain, (as
2y+1 = e^(x^2 - 1)e^(ln3)
:. 2y = 3e^(x^2 - 1) - 1
:. y = 3/2e^(x^2 - 1) - 1/2