Solution of differential equation x dy - y dx = 0 respresents : Option 1: family of straight lines passing through origin Option 2: family of parabolas whose vertex is at origin Option 3: family of circles whose centre is at origin Option 4: family of straight lines passing through (1, 1)

Answer: family of straight lines passing through origin. Solution: x dy = y dx dy/y = dx/x. Integrating: y = x + c, so y = kx. This represents a family of straight lines passing through the origin.

Calculus

Differential Equations

Easy

core

Solution of differential equation $x dy - y dx = 0$ respresents :

family of straight lines passing through origin

family of parabolas whose vertex is at origin

family of circles whose centre is at origin

family of straight lines passing through (1, 1)

✅ Correct Option: 1

$x\,dy = y\,dx \Rightarrow \frac{dy}{y} = \frac{dx}{x}$ . Integrating: $\ln y = \ln x + c$ , so $y = kx$ . This represents a family of straight lines passing through the origin.

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The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x^2+xy+y^2}{x^2}$ for $x>0, y>0$ at $x=1, y=2$ is

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Match List-I with List-II

List-I (Differential Equation)	List-II (Degree)
(A) $\left(\dfrac{dy}{dx}\right)^3+\left(\dfrac{d^2y}{dx^2}\right)^2=0$	(I) Not defined
(B) $\left(\dfrac{d^2y}{dx^2}\right)^3+\left(\dfrac{dy}{dx}\right)^2+1=0$	(II) 2
(C) $\sqrt{1+\dfrac{d^2y}{dx^2}}=\dfrac{dy}{dx}+x$	(III) 1
(D) $\dfrac{d^2y}{dx^2}+e^{\frac{dy}{dx}}=0$	(IV) 3

Choose the correct answer from the options given below:

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The sum of the order and degree of the differential equation representing the family of curves $y = mx + m^4$ , where m is arbitrary constant, is

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Solution of differential equation $\frac{dy}{dx} = \frac{1+x^2}{1+y^2}$ is

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The integrating factor of the differential equation, $x^2 \frac{dy}{dx} + xy = log_e x$ is equal to

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The general solution of the differential equation $(x+1)\dfrac{dy}{dx}=2e^{-y}-1$ is:

(where C is an arbitrary constant)

Solution of differential equation xdy−ydx=0x dy - y dx = 0xdy−ydx=0 respresents :

Solution

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Solution of differential equation $x dy - y dx = 0$ respresents :