Question

help needed to solve this linear differential eq : (given below)

Answer 1

\[\frac{ dy }{ dx } + y \cos x = e^{\sin x} \cos x\]

Answer 2

i took P = cos x and got the integrating factor as \[e^{\sin x} \]
the solution would be \[y. e^{\sin x} = \int\limits e^{\sin x} \cos x . e^{\sin x} dx\]

Answer 3

but i cannot integrate the Right hand side :(

Answer 4

\[\large e^{\sin x}y=\int\left(e^{\sin x}\right)^2\cos x~dx\]
Let \(u=\sin x\), then \(du=\cos x~dx\).
\[\large \begin{align*}e^{\sin x}y&=\int\left(e^{u}\right)^2~du\\\\
&=\int e^{2u}~du\end{align*}\]