The solution of the differential equation xdy - ydx = 0 represents Option 1: straight line passing through origin Option 2: a circle whose centre is at origin Option 3: a rectangular hyperbola Option 4: parabola whose vertex is at origin

Answer: straight line passing through origin. Solution: The differential equation is xdy - ydx = 0 Move ydx to the right side: xdy = ydx Divide both sides by xy: dy/y = dx/x Integrate both sides: dy/y = dx/x, y, =, x, + C where C is the constant of integration., y, -, x, = C Using the logarithm property (a) - (b) = (a/b):, y/x, = C Taking exponential on both sides:, y/x, = e^C Since e^C is a positive constant, let e^C = k where k > 0: y/x = ± k Let ± k = m where m is any constant: y/x = m y = mx The equation y = mx represents a straight line with slope m and y-intercept 0. Therefore, the solution represents a straight line passing through the origin.

Calculus

Differential Equations

Easy

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The solution of the differential equation $xdy - ydx = 0$ represents

straight line passing through origin

a circle whose centre is at origin

a rectangular hyperbola

parabola whose vertex is at origin

✅ Correct Option: 1

The differential equation is $xdy - ydx = 0$

Move $ydx$ to the right side:

$xdy = ydx$

Divide both sides by $xy$ :

$\frac{dy}{y} = \frac{dx}{x}$

Integrate both sides:

$\int \frac{dy}{y} = \int \frac{dx}{x}$

$\ln|y| = \ln|x| + C$

where $C$ is the constant of integration.

$\ln|y| - \ln|x| = C$

Using the logarithm property $\ln(a) - \ln(b) = \ln(a/b)$ :

$\ln\left|\frac{y}{x}\right| = C$

Taking exponential on both sides:

$\left|\frac{y}{x}\right| = e^C$

Since $e^C$ is a positive constant, let $e^C = k$ where $k > 0$ :

$\frac{y}{x} = \pm k$

Let $\pm k = m$ where $m$ is any constant:

$\frac{y}{x} = m$

$y = mx$

The equation $y = mx$ represents a straight line with slope $m$ and $y$ -intercept $0$ .

Therefore, the solution represents a straight line passing through the origin.

2025: 14 May Shift 1

Consider the differential equation, $x\frac{dy}{dx} = y(\log_e y - \log_e x + 1)$ , then which of the following are true?

(A) It is a linear differential equation

(B) It is a homogenous differential equation

(D) Its general solution is $\log_e\left(\frac{x}{y}\right) = Cy$ , where C is constant of integration

(E) If $y(1) = 1$ , then its particular solution is $y = x$

Choose the correct answer from the options given below:

2025: 15 May Shift 2

Consider the differential equation $xdy = (x + y) dx$ . Which of the following are true?

(A) It is a homogenous differential equation

(B) It is a differential equation of order 2

(D) Integrating factor of differential equation is $\frac{1}{x}$

(E) Degree of the differential equation is not defined

Choose the correct answer from the options given below:

2026: 19 May Shift 1

The sum of the order and degree of the differential equation: $\left(x + \frac{dy}{dx}\right)^2 = \frac{dy}{dx} + 1$ is

2023: 22 May Shift 3

The equation of curve whose slope is given by $\frac{dy}{dx} = x$ and which passes through $\left(1, \frac{5}{2}\right)$ is :

2025: 3 June Shift 2

The number of arbitrary constants in the particular solution of a differential equation of order 4 and degree 3 is

2023: 23 May Shift 3

In the context of differential equation

Match List I with List II

LIST I	LIST II
A. $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$	I. Not a differential equation
B. $x^2 \frac{dy}{dx} = x^2 - 2y^2 + xy$	II. Linear first order
C. $\sin x + y = \cos(x+y)$	III. Variable separable
D. $(x+y)\frac{dy}{dx} = 1$	IV. Homogenous

Choose the correct answer from the options given below:

The solution of the differential equation xdy−ydx=0xdy - ydx = 0xdy−ydx=0 represents

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The solution of the differential equation $xdy - ydx = 0$ represents