How to solve the following inhomogeneous differential equation? under the initial condition y (0) = y0. Let λ and μ be positive real parameters.

Question

#y'(x) = λy (x) + μe^(λx)#

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Answer 1 · 2018-01-06T18:02:42

See below.

This is a linear non-homogeneous differential equation. The general solution can be made as the sum

#y = y_h + y_p#

with

#y'_h-lambda y_h = 0# and
#y'_p-lambda y_p = mu e^(lambda x)#

The solution #y_h# is easily obtained and is

#y_h = C e^(lambda x)#

Now supposing that #y_p = C(x) e^(lambda x)# we have

#C'e^(lambda x)+lambda C e^(lambda x)-lambda C e^(lambda x) = mu e^(lambda x)# or simplifying

#C' = mu rArr C = mu x+C_0# and finally

#y = (mu x + C_0) e^(lambda x)# with the initial condition

#y(0) = C_0= y_0# so finally

#y = (mu x + y_0)e^(lambda x)#