List of top Mathematics Questions on Differential equations asked in UP Board XII

Find the general solution of the differential equation \(x\cos\left(\dfrac yx\right)\dfrac{dy}{dx} = y\cos\left(\dfrac yx\right)+x\).

Find the order and degree of the differential equation \(\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + \sin\left(\dfrac{dy}{dx}\right) + 1 = 0\).

\(x^2\,dy+\dfrac12(xy+y^2)\,dx=0\); find the particular solution, given that \(y=1\) when \(x=1\).

Solve the differential equation \(y\,dx+(x-y^2)\,dy=0\).

Verify that the function \(y=a\cos x+b\sin x\), in which \(a,b\in R\), is a solution of the differential equation \(\dfrac{d^2y}{dx^2}+y=0\).

The order of the differential equation \(x\left(\dfrac{d^3y}{dx^3}\right)^2+\sin x\left(\dfrac{d^2y}{dx^2}\right)^3+\left(\dfrac{dy}{dx}\right)^4+y^5=0\) is

Find the general solution of \(\dfrac{dy}{dx}=(1+x^2)(1+y^2)\).

Solve the differential equation \((x-y)\dfrac{dy}{dx}=x+2y\).

If \(y=Ae^{mx}+Be^{nx}\), then show that \(\dfrac{d^2y}{dx^2}-(m+n)\dfrac{dy}{dx}+mny=0\).

Solve the differential equation: \((1+x^2)\,dy+2xy\,dx=\cot x\,dx\).

Find the general solution of the differential equation \(\dfrac{dy}{dx}=\dfrac{1+y^2}{1+x^2}\).

If \(y=5\cos x-3\sin x\), then prove that \(\dfrac{d^2y}{dx^2}+y=0\).

Find the general solution of the differential equation \(\dfrac{dy}{dx}=\dfrac{1+y^2}{1+x^2}\).

Find the general solution of the differential equation \(\log\left(\dfrac{dy}{dx}\right)=3x+4y\).

Find the integrating factor of the differential equation \(x\dfrac{dy}{dx}+2y=x^2\ (x\neq0)\).

Show that the general solution of the differential equation \(\dfrac{dy}{dx}+\dfrac{y^{2}+y+1}{x^{2}+x+1}=0\) is \(x+y+1=A(1-x-y-2xy)\), in which \(A\) is a parameter.

Solve the differential equation \((x\,dy-y\,dx)\,y\sin\!\left(\dfrac{y}{x}\right)=(y\,dx+x\,dy)\,x\cos\!\left(\dfrac{y}{x}\right)\).

If \(y=Ae^{mx}+Be^{nx}\), then show that \(\dfrac{d^{2}y}{dx^{2}}-(m+n)\dfrac{dy}{dx}+mny=0\).

Find the particular solution of the differential equation \(\dfrac{dy}{dx}+y\cot x=4x\csc x\), \((x\neq0)\), given that \(y=0\) when \(x=\dfrac{\pi}{2}\).

Prove that the function \(y=Ae^{3x}\cos4x+Be^{3x}\sin4x\), where \(A\), \(B\) are arbitrary constants, is the solution of the differential equation \(\dfrac{d^2y}{dx^2}-6\dfrac{dy}{dx}+25y=0\).

If \(\dfrac{dy}{dx}=e^{x+y}\), then its solution is: