Differential Equations
26 Board Maths previous year questions on Differential Equations — options free on every question; 3 include the answer & explanation free, the rest unlock with PYQ Pass.
The solution of the differential equation \(\frac{dy}{dx}=\frac{y}{x}\) is
\(y=cx\)
The given differential equation is:
\(\frac{dy}{dx}=\frac{y}{x}.....(I)\)
Rearrange the tems of the equation \(\left(I\right)\)
\(\frac{dy}{y}=\frac{dx}{x}.....(II)\)
Integrate on both sides of equation \((II)\) and calculate the solution of the given differential equation.
\(\int \frac{dy}{y}=\int \frac{dx}{x}\\ \log (y)=\log (x)+\log (c)\\ \log (y)=\log (cx)\\ y=cx\)
Hence, the solution of the given differential equation is \(y=cx\).
Let \(y=y(x)\) be the solution of the differential equation \({x}^{4}dy+\left(4{x}^{3}y+2\sin x\right)dx=0\), \(x>0,y\left(\frac{\pi }{2}\right)=0\). Then \({\pi }^{4}y\left(\frac{\pi }{3}\right)\) is equal to:
[JEE Main 2026, 23 Jan (Shift 2)]
81
Given: \(\left(x^4 d y+4 x^3 y d x\right)=-2 \sin x d x\)
\(\Rightarrow \int d\left(x^4 y\right)=\int-2 \sin x d x\)
\(\Rightarrow x^4 y=2 \cos x+c\)
As \(y\left(\frac{\pi}{2}\right)=0\)
So, \(c=0\)
Now, \(\left(\frac{\pi}{3}\right)^4 y\left(\frac{\pi}{3}\right)=2 \cos \frac{\pi}{3}\)
\(\Rightarrow \pi^4 y\left(\frac{\pi}{3}\right)=81\)
The integrating factor of the differential equation \(\left(1-x^2\right) \frac{d y}{d x}+x y=a x\), \(-1
\(\frac{1}{\sqrt{1-\mathrm{x}^2}}\)
\(\left(1-x^2\right) \frac{d y}{d x}+x y=a x\)
\(\Rightarrow \frac{d y}{d x}+\frac{x}{1-x^2} y=\frac{a x}{1-x^2}\)
\(P(x)=\frac{x}{1-x^2}\)
\(\int P(x) d x=\int \frac{x}{1-x^2} d x\)
Let \(u=1-x^2, d u=-2 x d x\)
\(\int \frac{x}{1-x^2} d x=-\frac{1}{2} \int \frac{d u}{u}=-\frac{1}{2} \ln \left(1-x^2\right)\)
I.F. \(=e^{\int P d x}=e^{-\frac{1}{2} \ln \left(1-x^2\right)}=\left(1-x^2\right)^{-1 / 2}\)
The differential equation \(\frac{d y}{d x}=F(x, y)\) will not be a homogeneous differential equation, if \(\mathrm{F}(\mathrm{x}, \mathrm{y})\) is :
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Let \(f:\left[1,∞\right)\to ℝ\) be a differentiable function. If \(6{\int }_{1}^{x}f\left(t\right)dt=3xf\left(x\right)+{x}^{3}−4\) for all \(x\geq 1\), then the value of \(f(2)–f(3)\) is
[JEE Main 2026, 22 Jan (Shift 1)]
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The degree of the differential equation \(\left(y^{\prime \prime}\right)^2+\left(y^{\prime}\right)^3=x \sin \left(y^{\prime}\right)\) is :
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The order and degree of the differential equation
\(xy\left(\frac{{d}^{2}y}{d{x}^{2}}\right)+x{\left(\frac{dy}{dx}\right)}^{2}-y\left(\frac{dy}{dx}\right)=0\) is
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The integrating factor of the differential equation \(\frac{dy}{dx}+2y={e}^{3x}\) is
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Let the solution curve of the differential equation \(xdy−ydx=\sqrt{{x}^{2}+{y}^{2}}dx,x>0\), \(y(1)=0,\) be \(y=y(x)\). Then \(y(3)\) is equal to
[JEE Main 2026, 22 Jan (Shift 1)]
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The integrating factor of the differential equation \(\frac{dy}{dx}+2y={e}^{3x}\) is
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The integrating factor of the differential equation \(\left(1-x^2\right) \frac{d y}{d x}+x y=a x\), \(-1
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The order of the differential equation \(\frac{{\mathrm{d}}^{4}\mathrm{y}}{{\mathrm{dx}}^{4}}-\sin \left(\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}\right)=5\) is :
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The integrating factor of the differential equation
\(\frac{dy}{dx}+2y=\sin x\) is
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If \(p\) and \(q\) are respectively the order and degree of the differential equation \(\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^3=0\), then \((p-q)\) is
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The integrating factor of the differential equation
\(\frac{dy}{dx}+2y=\sin x\) is
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The order and degree of the differential equation \(\left[1+\left(\frac{d y}{d x}\right)^2\right]^3=\frac{d^2 y}{d x^2}\) respectively are :
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The order and degree of the differential equation
\(xy\left(\frac{{d}^{2}y}{d{x}^{2}}\right)+x{\left(\frac{dy}{dx}\right)}^{2}-y\left(\frac{dy}{dx}\right)=0\) is
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The solution of the differential equation \(\frac{dy}{dx}=\frac{y}{x}\) is
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If \(y=y(x)\) satisfies the differential equation \(16\left(\sqrt{9x+x\sqrt{x}}\right)\left(4+\sqrt{9+\sqrt{x}}\right)\) \(\cos y\text{ }dy=(1+2\sin y)\)\(dx,\text{ }x>0\) and \(y(256)=\frac{\pi }{2},\text{ }y(49)=\alpha ,\) then \(2\sin \alpha\) is equal to
[JEE Main 2026, 22 Jan (Shift 2)]
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The order of the differential equation \(\frac{{\mathrm{d}}^{4}\mathrm{y}}{{\mathrm{dx}}^{4}}-\sin \left(\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}\right)=5\) is :
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The differential equation \(\frac{d y}{d x}=F(x, y)\) will not be a homogeneous differential equation, if \(\mathrm{F}(\mathrm{x}, \mathrm{y})\) is :
Options are free to see. Unlock the correct answer and full explanation with Pass.
The order and degree of the differential equation \(\left[1+\left(\frac{d y}{d x}\right)^2\right]^3=\frac{d^2 y}{d x^2}\) respectively are :
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The degree of the differential equation \(\left(y^{\prime \prime}\right)^2+\left(y^{\prime}\right)^3=x \sin \left(y^{\prime}\right)\) is :
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If \(p\) and \(q\) are respectively the order and degree of the differential equation \(\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^3=0\), then \((p-q)\) is
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The number of arbitrary constants in the general solution of a differential equation of fourth order are :
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The anti derivative of \(\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)\) with respect to x-
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