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 #
