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

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

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

Answers (1)

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)
\(ue^(-5v)= (ve^(-10v))/-10-1/100 e^(-10v)+c\)
\(u= (ve^(-10v))/(-10 e^(-5v))-1/100 e^(-10v)/e^(-5v)+c/e^(-5v)\)
\(u= ve^(-5v)/-10-1/100 e^(-5v)+ce^(5v)\)
0

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