jee-main 2024 Q77

jee-main · India · session2_06apr_shift1 First order differential equations (integrating factor)
Let $y = y ( x )$ be the solution of the differential equation $\left( 1 + x ^ { 2 } \right) \frac { d y } { d x } + y = e ^ { \tan ^ { - 1 } x } , y ( 1 ) = 0$. Then $y ( 0 )$ is
(1) $\frac { 1 } { 2 } \left( e ^ { \pi / 2 } - 1 \right)$
(2) $\frac { 1 } { 2 } \left( 1 - e ^ { \pi / 2 } \right)$
(3) $\frac { 1 } { 4 } \left( 1 - e ^ { \pi / 2 } \right)$
(4) $\frac { 1 } { 4 } \left( e ^ { \pi / 2 } - 1 \right)$
Let $y = y ( x )$ be the solution of the differential equation $\left( 1 + x ^ { 2 } \right) \frac { d y } { d x } + y = e ^ { \tan ^ { - 1 } x } , y ( 1 ) = 0$. Then $y ( 0 )$ is\\
(1) $\frac { 1 } { 2 } \left( e ^ { \pi / 2 } - 1 \right)$\\
(2) $\frac { 1 } { 2 } \left( 1 - e ^ { \pi / 2 } \right)$\\
(3) $\frac { 1 } { 4 } \left( 1 - e ^ { \pi / 2 } \right)$\\
(4) $\frac { 1 } { 4 } \left( e ^ { \pi / 2 } - 1 \right)$