By the method of separation of variables we suppose that u is of the form u(t,x)=ϕ(t)ψ(x) which gives us :
ϕ′(t)ψ(x)+ϕ(t)ψ′(x)+2etϕ(t)ψ(x)=0
dividing both parts by u(t,x)=ϕ(t)ψ(x) gives us
ϕ′(t)/ϕ(t)+2et+ψ′(x)/ψ(x)=0
ϕ′(t)/ϕ(t)+2et=−ψ′(x)/ψ(x)
As the left part is a function of t and the right is the function of x the equality gives us
{ϕ′/ϕ+2et=λ−ψ′/ψ=λ with λ∈R a constant.
The second equation easily gives us ψλ(x)=Ae−λx, A a constant
The first one gives us
lnϕ(t)−lnϕ(0)=λt−2et+2
ϕλ(t)=Beλt−2et , B a constant
Combining two solutions gives us
uλ(t,x)=Ceλ(t−x)e−2et
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