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1
JEE Main 2021 (Online) 27th July Evening Shift
Numerical
+4
-1
Let y = y(x) be the solution of the differential equation dy = e$$\alpha$$x + y dx; $$\alpha$$ $$\in$$ N. If y(loge2) = loge2 and y(0) = loge$$\left( {{1 \over 2}} \right)$$, then the value of $$\alpha$$ is equal to _____________.
2
JEE Main 2021 (Online) 27th July Morning Shift
Numerical
+4
-1
Let $$F:[3,5] \to R$$ be a twice differentiable function on (3, 5) such that $$F(x) = {e^{ - x}}\int\limits_3^x {(3{t^2} + 2t + 4F'(t))dt}$$. If $$F'(4) = {{\alpha {e^\beta } - 224} \over {{{({e^\beta } - 4)}^2}}}$$, then $$\alpha$$ + $$\beta$$ is equal to _______________.
3
JEE Main 2021 (Online) 27th July Morning Shift
Numerical
+4
-1
If $$y = y(x),y \in \left[ {0,{\pi \over 2}} \right)$$ is the solution of the differential equation $$\sec y{{dy} \over {dx}} - \sin (x + y) - \sin (x - y) = 0$$, with y(0) = 0, then $$5y'\left( {{\pi \over 2}} \right)$$ is equal to ______________.
Let y = y(x) be solution of the following differential equation $${e^y}{{dy} \over {dx}} - 2{e^y}\sin x + \sin x{\cos ^2}x = 0,y\left( {{\pi \over 2}} \right) = 0$$ If $$y(0) = {\log _e}(\alpha + \beta {e^{ - 2}})$$, then $$4(\alpha + \beta )$$ is equal to ______________.