JEE Main 2024 (Online) 5th April Morning Shift | Differential Equations Question 14 | Mathematics

If $$y=y(x)$$ is the solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=\sin (2 x), y(0)=\frac{3}{4}$$, then $$y\left(\frac{\pi}{8}\right)$$ is equal to :

$$\mathrm{e}^{-\pi / 8}$$

$$\mathrm{e}^{\pi / 4}$$

$$\mathrm{e}^{-\pi / 4}$$

$$\mathrm{e}^{\pi / 8}$$

JEE Main 2024 (Online) 4th April Evening Shift

MCQ (Single Correct Answer)

-1

Let $$y=y(x)$$ be the solution of the differential equation $$(x^2+4)^2 d y+(2 x^3 y+8 x y-2) d x=0$$. If $$y(0)=0$$, then $$y(2)$$ is equal to

$$2 \pi$$

$$\frac{\pi}{8}$$

$$\frac{\pi}{16}$$

$$\frac{\pi}{32}$$

JEE Main 2024 (Online) 4th April Morning Shift

MCQ (Single Correct Answer)

-1

If the solution $$y=y(x)$$ of the differential equation $$(x^4+2 x^3+3 x^2+2 x+2) \mathrm{d} y-(2 x^2+2 x+3) \mathrm{d} x=0$$ satisfies $$y(-1)=-\frac{\pi}{4}$$, then $$y(0)$$ is equal to :

$$-\frac{\pi}{12}$$

$$\frac{\pi}{2}$$

$$\frac{\pi}{4}$$

JEE Main 2024 (Online) 1st February Evening Shift

MCQ (Single Correct Answer)

-1

Let $\alpha$ be a non-zero real number. Suppose $f: \mathbf{R} \rightarrow \mathbf{R}$ is a differentiable function such that $f(0)=2$ and $\lim\limits_{x \rightarrow-\infty} f(x)=1$. If $f^{\prime}(x)=\alpha f(x)+3$, for all $x \in \mathbf{R}$, then $f\left(-\log _{\mathrm{e}} 2\right)$ is equal to :