PYQ Vault
EbooksTutor

Homogeneous equation x f(y/x) dy = (y f(y/x) +- x) dx with f a trigonometric function

This type of question, asked 3 times: 2021 Paper 11, 29 Jan 2024, 5 Apr 2026 S1. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 11

    51
    Let y=y(x)y=y(x) be the solution of the differential equation xtan(yx)dy=(ytan(yx)−x)dxxtan\left( \frac{y}{x} \right)dy=\left( ytan\left( \frac{y}{x} \right)-x \right)dx, −1≤x≤1,y(12)=π6- 1 \leq x \leq 1,y\left( \frac{1}{2} \right)=\frac{\pi}{6}. Then the area of the region bounded by the curves x=0,x=12x= 0,x=\frac{1}{\sqrt{2}} and y=y(x)y=y(x) in the upper half plane is:

    Tap an option to check your answer.

  2. Question 2

    29 Jan 2024

    162
    If sin⁡(yx)=log⁡e∣x∣+α2\sin\left( \frac{y}{x} \right)=\log_{e}|x| +\frac{\alpha}{2} is the solution of the differential equation xcos(yx)dydx=ycos(yx)+xxcos\left( \frac{y}{x} \right)\frac{dy}{dx}= ycos\left( \frac{y}{x} \right)+ x and y(1)=π3y(1) =\frac{\pi}{3}, then α2\alpha^{2} is equal to

    Tap an option to check your answer.

  3. Question 3

    5 Apr 2026 S1

    75
    Let y=y(x)y = y(x) be the solution of the differential equation x sin⁡(yx)dy=(ysin⁡(yx)−x)dx,y(1)=π2x\ \sin\left( \frac{y}{x} \right)dy = \left( y\sin\left( \frac{y}{x} \right) - x \right)dx,y(1) = \frac{\pi}{2} and let α=cos⁡(y(e12)e12)\alpha = \cos\left( \frac{y\left( e^{12} \right)}{e^{12}} \right). Then the number of integral value of pp, for which the equation x2+y2−2px+2py+α+2=0x^{2} + y^{2} - 2px + 2py + \alpha + 2 = 0 represents a circle of radius r≤6r \leq 6, is ____\_\_\_\_ .