# 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 #