Calculus

Differential Equations

Medium

core

If $\sin y=x \sin (a+y)$ , then $\frac{d y}{d x}$ is :

$\frac{\sin ^{2} a}{\sin (a+y)}$

$\frac{\sin (a+y)}{\sin ^{2} a}$

$\frac{\sin (a+y)}{\sin a}$

$\frac{\sin ^{2}(a+y)}{\sin a}$

✅ Correct Option: 4

Starting from:

$\sin y = x \sin(a + y)$

Let's solve for $x$ first, as this makes differentiation much easier:

$x = \dfrac{\sin y}{\sin(a + y)}$

We'll find $\dfrac{dx}{dy}$ and then flip it to get $\dfrac{dy}{dx} = \dfrac{1}{\frac{dx}{dy}}$ .

Differentiating $x$ with respect to $y$ using the quotient rule:

$\dfrac{dx}{dy} = \dfrac{\sin(a+y) \cdot \cos y - \sin y \cdot \cos(a+y)}{\sin^2(a+y)}$

The numerator matches the sine subtraction identity: $\sin A \cos B - \cos A \sin B = \sin(A - B)$

With $A = (a+y)$ and $B = y$ :

$\sin(a+y)\cos y - \sin y \cos(a+y) = \sin((a+y) - y) = \sin a$

So:

$\dfrac{dx}{dy} = \dfrac{\sin a}{\sin^2(a+y)}$

Flipping to get $\dfrac{dy}{dx}$ :

$\dfrac{dy}{dx} = \dfrac{1}{\frac{dx}{dy}} = \dfrac{\sin^2(a+y)}{\sin a}$

$\boxed{\dfrac{dy}{dx} = \dfrac{\sin^2(a+y)}{\sin a}}$

2025: 22 May Shift 2

Match List-I with List-II

List-I	List-II
Differential Equation	Order and Degree
(A) $\left(\frac{d^2y}{dx^2}\right)^2 = e^x\left(\frac{dy}{dx}\right)^4 + 1 = 0$	(I) order = 1 and degree = 2
(B) $\left(\frac{dy}{dx}\right)^2 + xy = 0$	(II) order = 2 and degree = 1
(C) $\left(1 + \frac{dy}{dx}\right)^{3/2} = 4\left(\frac{d^2y}{dx^2}\right)^2$	(III) order = 2 and degree = 2
(D) $\sqrt\frac{d^2y}{dx^2} + 1 = \frac{dy}{dx}$	(IV) order = 2 and degree = 4

Choose the correct answer from the options given below:

2025: 14 May Shift 1

The solution of the differential equation $log_e\left(\frac{dy}{dx}\right) = 3x + 4y$ is given by

2025: 30 May Shift 1

Which of the following are first order linear differential equations?

(A) $\frac{dx}{dy} + P_1(y)x = Q_1(y)$ : $P_1(y)$ and $Q_1(y)$ are functions of y or constant functions

(B) $\frac{dy}{dx} + P_2(x)y = Q_2(x)$ : $P_2(x)$ and $Q_2(x)$ are functions of x or constant functions

(D) $(1 + x^2)\frac{dy}{dx} - 2xy = x^2 + 3$

Choose the correct answer from the options given below:

2025: 13 May Shift 1

The general solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = ax + by$ is

2026: 22 May Shift 2

The general solution of the differential equation $\frac{dy}{dx} = \frac{3e^{2x}+3e^{4x}}{e^x+e^{-x}}$ is

2022: 25 May Shift 1

The degree of the differential equation

$\left(1 + \frac{dy}{dx}\right)^4 = \left(\frac{d^2y}{dx^2}\right)^2$ is:

If sin⁡y=xsin⁡(a+y)\sin y=x \sin (a+y)siny=xsin(a+y), then dydx\frac{d y}{d x}dxdy​ is :

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

If $\sin y=x \sin (a+y)$ , then $\frac{d y}{d x}$ is :