IAT (IISER) 2025 | Differential Equations Question 1 | Mathematics

Consider the differential equation $\cos (y) \frac{d y}{d x}+\frac{1}{x} \sin (y)=x, \quad(x>0)$; given that, $y=\frac{\pi}{2}$ at $x=\sqrt{3}$. Which one of the following is the value of $y$ at $x=\sqrt{\frac{3}{2}}$ ?

$\frac{\pi}{6}$

$\frac{\pi}{3}$

$\frac{\pi}{2}$

$\frac{\pi}{4}$

IAT (IISER) 2023

MCQ (Single Correct Answer)

-1

Which of the following differential equations has $y=e^x$ as one of its particular solutions?

$y \frac{d^2 y}{d x^2}+e^x \frac{d y}{d x}+y^2=e^{2 x}$

$y \frac{d^2 y}{d x^2}-e^x \frac{d y}{d x}+y^2=e^{2 x}$

$y \frac{d^2 y}{d x^2}-e^x \frac{d y}{d x}+y^2=e^x$

$y \frac{d^2 y}{d x^2}+e^x \frac{d y}{d x}+y^2=e^x$

IAT (IISER) 2022

MCQ (Single Correct Answer)

-1

For arbitrary constants $\alpha, \beta$, the differential equation representing the family of curves $y=$ $(\alpha x+\beta) e^x$ is

$y^{\prime \prime}-2 y^{\prime}+y=0$

$y^{\prime \prime}-y^{\prime}+y=0$

$y^{\prime \prime}-2 y^{\prime}-y=0$

$y^{\prime \prime}-y^{\prime}-y=0$

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