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Question
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
Options
\[y e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]
\[y e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]
\[x e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]
\[x e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]
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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 = P1
Q = Q1
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
\[x e^{\int P_1 dy} = \int\left\{ Q_1 e^{\int P_1 dy} \right\}dy + C\]
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