Differential Equations17 questions

Differential Equations — JEE Maths Practice Questions & Solutions

17 questions on Differential Equations with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

medium
The degree of the differential equation satisfying the relation
1+x2+1+y2=λ(x1+y2−y1+x2)\sqrt{1+x^2} + \sqrt{1+y^2} = \lambda\big(x\sqrt{1+y^2} - y\sqrt{1+x^2}\big)
 is:
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medium
If pp and qq are the order and degree of the differential equation y2(d2ydx2)2+3x(dydx)1/3+x2y2=sin⁡xy^2\left(\dfrac{d^2y}{dx^2}\right)^2 + 3x\left(\dfrac{dy}{dx}\right)^{1/3} + x^2y^2 = \sin x, then:
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medium
A curve, whose concavity is directly proportional to the logarithm of its xx-coordinate at any point of the curve, is given by:
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medium
The solution of the differential equation (x2sin⁡3y−y2cos⁡x) dx+(x3cos⁡ysin⁡2y−2ysin⁡x) dy=0(x^2\sin^3 y - y^2\cos x)\,dx + (x^3\cos y\sin^2 y - 2y\sin x)\,dy = 0 is:
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medium
The solution of the differential equation dxdy−xlog⁡x1+log⁡x=ey1+log⁡x\dfrac{dx}{dy} - \dfrac{x\log x}{1+\log x} = \dfrac{e^y}{1+\log x}, if y(1)=0y(1) = 0, is:
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medium
If a curve y=f(x)y = f(x) satisfies y′′x2+x(y′)2=2xy′y''x^2 + x(y')^2 = 2xy', with f(0)=0f(0) = 0 and f′(1)=1f'(1) = 1, then f(x)f(x) is:
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medium
A continuous function f:R→Rf: R \to R satisfies f(x)=(1+x2)[1+∫0xf2(t)1+t2 dt]f(x) = (1+x^2)\left[1 + \displaystyle\int_0^x \dfrac{f^2(t)}{1+t^2}\,dt\right]. Then the value of f(−2)f(-2) is:
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medium
The solution of the differential equation x dx+y dy+x dy−y dxx2+y2=0x\,dx + y\,dy + \dfrac{x\,dy - y\,dx}{x^2 + y^2} = 0 is:
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medium
The solution of {y(1+x−1)+sin⁡y} dx+(x+ln⁡x+xcos⁡y) dy=0\big\{y(1 + x^{-1}) + \sin y\big\}\,dx + (x + \ln x + x\cos y)\,dy = 0 is:
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medium
The solution of the differential equation dydx=sin⁡y+xsin⁡2y−xcos⁡y\dfrac{dy}{dx} = \dfrac{\sin y + x}{\sin 2y - x\cos y} is:
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medium
The primitive of the differential equation (2xy4ey+2xy3+y) dx+(x2y4ey−x2y2−3x) dy=0(2xy^4e^y + 2xy^3 + y)\,dx + (x^2y^4e^y - x^2y^2 - 3x)\,dy = 0 is:
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medium
A function y=f(x)y = f(x) satisfies f′(x)sin⁡x+f(x)cos⁡x=1f'(x)\sin x + f(x)\cos x = 1, with f(x)f(x) bounded as x→0x \to 0. If I=∫0π/2f(x) dxI = \displaystyle\int_0^{\pi/2} f(x)\,dx, then:
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easy
The order of the differential equation corresponding to y=c1cos⁡2x+c2cos⁡2x+c3sin⁡2x+c4y = c_1\cos 2x + c_2\cos^2 x + c_3\sin^2 x + c_4 is:
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easy
The differential equation of all parabolas with axis parallel to the yy-axis is:
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easy
Let x dydx−y=x2(xex+ex−1)\dfrac{x\,dy}{dx} - y = x^2(xe^x + e^x - 1) for all x∈R−{0}x \in R - \{0\} with y(1)=e−1y(1) = e - 1. If y(2)=k y(1)(y(1)+2)y(2) = k\,y(1)\big(y(1)+2\big), then the value of kk is:
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easy
The equation dydx=x2+y2+12xy\dfrac{dy}{dx} = \dfrac{x^2 + y^2 + 1}{2xy}, with y(1)=1y(1) = 1, is the differential equation of:
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easy
The differential equation ey dx+(eyx+2y) dy=0e^y\,dx + (e^y x + 2y)\,dy = 0 has the particular solution y(0)=1y(0) = 1. The value of xx when y=0y = 0 is:
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