#### Question

The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is

\[y e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]

\[y e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]

\[x e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]

\[x e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]

#### Solution

\[x e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]

We have,

\[\frac{dx}{dy} + P_1 x = Q_1\]

Comparing with the equation \[\frac{dx}{dy} + Px = Q\], we get

P = P_{1}

Q = Q_{1}

The general solution of the equation \[\frac{dx}{dy} + Px = Q\] is given by \[x e^{\int Pdy} = \int\left\{ Q e^{\int Pdy} \right\}dy + C\] ...(1)

Putting the value of P and Q in (1), we get

\[x e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]