The general solution of a differential equation of the type $\frac{{dx}}{{dy}} + {{\rm{P}}_1}x = {{\rm{Q}}_1}$, is
A. $y{e^{\int {{\rm{P}}_1^{}dy} }} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{{\rm{P}}_1}} dy}}} \right)} dy + {\rm{C}}$
B. $y \cdot {e^{\int {{\rm{P}}_1^{}dx} }} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{{\rm{P}}_1}} dx}}} \right)} dx + {\rm{C}}$
C. $x{e^{\int {{\rm{P}}_1^{}dy} }} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{\rm{P}}_1^{}} dy}}} \right)} dy + {\rm{C}}$
D. $x{e^{\int {{\rm{P}}_1^{}} dx}} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{\rm{P}}_1^{}} dx}}} \right)} dx + {\rm{C}}$
The general solution of a differential equation of the type $\frac{{dx}}{{dy}} + {{\rm{P}}_1}x = {{\rm{Q}}_1}$, is
A. $y{e^{\int {{\rm{P}}_1^{}dy} }} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{{\rm{P}}_1}} dy}}} \right)} dy + {\rm{C}}$
B. $y \cdot {e^{\int {{\rm{P}}_1^{}dx} }} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{{\rm{P}}_1}} dx}}} \right)} dx + {\rm{C}}$
C. $x{e^{\int {{\rm{P}}_1^{}dy} }} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{\rm{P}}_1^{}} dy}}} \right)} dy + {\rm{C}}$
D. $x{e^{\int {{\rm{P}}_1^{}} dx}} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{\rm{P}}_1^{}} dx}}} \right)} dx + {\rm{C}}$
Official Solution
Option C is correct
The integrating factor of the given differential equation $\frac{{dx}}{{dy}} + {{\rm{P}}_1}x = {{\rm{Q}}_1}$ is ${e^{\int {{P_1}} dy}}$.
The general solution of the differential equation is given by, $x({\rm{I}}.{\rm{F}}.) = \int {({\rm{Q}} \times {\rm{I}}.{\rm{F}}.)} dy + {\rm{C}}$
$\Rightarrow$ $x \cdot {e^{\int {{P_1}dy} }} = \int {\left( {{Q_1}{e^{\int {{P_1}_{dy}} }}} \right)} dy + {\rm{C}}$
Hence, the correct answer is C.
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