# Solve differential equation u'-5u=ve^(-5v)

Question
Solve differential equation $$u'-5u=ve^(-5v)$$

2021-03-06
$$(du)/(dv)-5u= ve^(-5v)$$
$$dy/dx+Py=Q$$
$$I.F.= e^(int Pdx)$$
$$= e^(- int 5dv)$$
$$= e^(-5 int dv)$$
$$= e^(-5v)$$
$$u(I.F.)= int ve^(-5v)(I.F.)dv+c$$
$$ue^(-5v)= int ve^(-5v) e^(-5v)dv+c$$
$$ue^(-5v)= int ve^(-10v)dv+c$$ (1)
Now find $$int ve^(-10v)dv$$ (by parts)
$$= v int e^(-10v)dv- int [d/(dv)(v) int e^(-10v)dv]dv$$
$$= v e^(-10v)/-10- int [e^(-10v)/-10]dv$$
$$= (ve^(-10v))/-10+1/10(e^(-10v)/-10)$$ (2)
from (1) and (2)
$$ue^(-5v)= (ve^(-10v))/-10-1/100 e^(-10v)+c$$
$$u= (ve^(-10v))/(-10 e^(-5v))-1/100 e^(-10v)/e^(-5v)+c/e^(-5v)$$
$$u= ve^(-5v)/-10-1/100 e^(-5v)+ce^(5v)$$

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