The general solution of a differential equation of the type dy + 1 x = 1 , is A. y e 1 dy = 1 e 1 dy ) dy + B. y 1 dx = 1 e 1 dx ) dx + C. x e 1 dy = 1 e 1 dy ) dy + D. x e 1 dx = 1 e 1 dx ) dx +

Option C is correct The integrating factor of the given differential equation dy + 1 x = 1 is e dy . The general solution of the differential equation is given by, x( . .) = . .) dy + x dy = e dy ) dy + Hence, the correct answer is C.

Differential Equations — Class 12 Maths Solution

ncert misc SA NCERT Misc.,Q.17,Page 421

Question

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. $x{e^{\int {{\rm{P}}_1^{}} dx}} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{\rm{P}}_1^{}} dx}}} \right)} dx + {\rm{C}}$

Step-by-step 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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