The general solution of the differintial equation : xdy/dx-ay=1(a≠0) is Option 1: y=-1/a+Cx^a:Where C is an arbitrary constant Option 2: y=-1/a+Cx^(-a):Where C is an arbitrary constant Option 3: y=-1/a+Cx:Where C is an arbitrary constant Option 4: y=-1/a+Cx^(-a-1):Where C is an arbitrary constant

Answer: y=-1/a+Cx^a:Where C is an arbitrary constant. Solution: Write it as dy/dx-a/xy=1/x, a linear equation with integrating factor x^(-a). Then d/dx(yx^(-a))=x^(-a-1), so yx^(-a)=-(x^(-a))/a+C. Hence y=-1/a+Cx^a.

Calculus

Differential Equations

Medium

core

The general solution of the differintial equation : $x\frac{dy}{dx}-ay=1(a\neq0)$ is

$y=-\frac{1}{a}+Cx^a$ :Where C is an arbitrary constant

$y=-\frac{1}{a}+Cx^{-a}$ :Where C is an arbitrary constant

$y=-\frac{1}{a}+Cx$ :Where C is an arbitrary constant

$y=-\frac{1}{a}+Cx^{-a-1}$ :Where C is an arbitrary constant

✅ Correct Option: 1

Write it as $\frac{dy}{dx}-\frac{a}{x}y=\frac{1}{x}$ , a linear equation with integrating factor $x^{-a}$ .

Then $\frac{d}{dx}(yx^{-a})=x^{-a-1}$ , so $yx^{-a}=-\frac{x^{-a}}{a}+C$ .

Hence $y=-\frac{1}{a}+Cx^{a}$ .

2025: 13 May Shift 2

The general solution of the differential equation $e^x dy + (y e^x + 2x)dx = 0$ is

2025: 16 May Shift 1

The particular solution of the differential equation x(1 + y²)dx - y(1 + x²)dy = 0, y(0) = 1, is:

2026: 22 May Shift 2

Match List-I with List-II

List-I: Linear differential equation	List-II: Integrating factor
(A) $x\frac{dy}{dx}-y=2x^3$	(I) $e^{-x}$
(B) $2x\frac{dy}{dx}+y=3x^2$	(II) $e^x$
(C) $\frac{dy}{dx}+y=\cos x-\sin x$	(III) $x^{1/2}$
(D) $\frac{dy}{dx}-y=e^x$	(IV) $1/x$

Choose the correct answer from the options given below:

2025: 30 May Shift 2

The solution of the differential equation $\frac{dy}{dx} = \sqrt\frac{{y}}{x}$ 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 :

2026: 29 May Shift 1

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:

The general solution of the differintial equation : xdydx−ay=1(a≠0)x\frac{dy}{dx}-ay=1(a\neq0)xdxdy​−ay=1(a=0) is

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The general solution of the differintial equation : $x\frac{dy}{dx}-ay=1(a\neq0)$ is