For each of the differential equations given below, indicate its order and degree (if defined).
(i) \(\frac{d^2y}{dx^2}+5x\left(\frac{dy}{dx}\right)^2-6y=\log x\)
(ii) \(\left(\frac{dy}{dx}\right)^3-4\left(\frac{dy}{dx}\right)^2+7y=\sin x\)
(iii) \(\frac{d^4y}{dx^4}-\sin\left(\frac{d^3y}{dx^3}\right)=0\)
(i) Order 2; Degree 1
(ii) Order 1; Degree 3
(iii) Order 4; Degree not defined
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(i) \(xy=ae^x+be^{-x}+x^2\) : \(x\frac{d^2y}{dx^2}+2\frac{dy}{dx}-xy+x^2-2=0\)
(ii) \(y=e^x(a\cos x+b\sin x)\) : \(\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0\)
(iii) \(y=x\sin 3x\) : \(\frac{d^2y}{dx^2}+9y-6\cos 3x=0\)
(iv) \(x^2=2y^2\log y\) : \((x^2+y^2)\frac{dy}{dx}-xy=0\)
Prove that \(x^2-y^2=c(x^2+y^2)^2\) is the general solution of differential equation \((x^3-3xy^2)\,dx=(y^3-3x^2y)\,dy\), where \(c\) is a parameter.
Find the general solution of the differential equation \(\frac{dy}{dx}+\sqrt{\frac{1-y^2}{1-x^2}}=0\).
\(\sin^{-1}y+\sin^{-1}x=C\)
Show that the general solution of the differential equation \(\frac{dy}{dx}+\frac{y^2+y+1}{x^2+x+1}=0\) is given by \((x+y+1)=A(1-x-y-2xy)\), where \(A\) is parameter.
Find the equation of the curve passing through the point \((0,\frac{\pi}{4})\) whose differential equation is \(\sin x\cos y\,dx+\cos x\sin y\,dy=0\).
\(\cos y=\frac{\sec x}{\sqrt{2}}\)
Find the particular solution of the differential equation \((1+e^{2x})\,dy+(1+y^2)e^x\,dx=0\), given that \(y=1\) when \(x=0\).
\(\tan^{-1}y+\tan^{-1}(e^x)=\frac{\pi}{2}\)
Solve the differential equation \(y e^{\frac{x}{y}}\,dx=(x e^{\frac{x}{y}}+y^2)\,dy\) \((y\ne 0)\).
\(e^{\frac{x}{y}}=y+C\)
Find a particular solution of the differential equation \((x-y)(dx+dy)=dx-dy\), given that \(y=-1\), when \(x=0\). (Hint: put \(x-y=t\))
\(\log|x-y|=x+y+1\)
Solve the differential equation \(\left[\frac{e^{-2\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right]\frac{dx}{dy}=1\) \((x\ne 0)\).
\(y e^{2\sqrt{x}}=(2\sqrt{x}+C)\)
Find a particular solution of the differential equation \(\frac{dy}{dx}+y\cot x=4x\cosec x\) \((x\ne 0)\), given that \(y=0\) when \(x=\frac{\pi}{2}\).
\(y\sin x=2x^2-\frac{\pi^2}{2}\) \((\sin x\ne 0)\)
Find a particular solution of the differential equation \((x+1)\frac{dy}{dx}=2e^{-y}-1\), given that \(y=0\) when \(x=0\).
\(y=\log\left|\frac{2x+1}{x+1}\right|,\ x\ne -1\)
The general solution of the differential equation \(\frac{y\,dx-x\,dy}{y}=0\) is
\(xy=C\)
\(x=Cy^2\)
\(y=Cx\)
\(y=Cx^2\)
The general solution of a differential equation of the type \(\frac{dx}{dy}+P_1x=Q_1\) is
\(y\,e^{\int P_1\,dy}=\int\left(Q_1e^{\int P_1\,dy}\right)dy+C\)
\(y\,e^{\int P_1\,dx}=\int\left(Q_1e^{\int P_1\,dx}\right)dx+C\)
\(x\,e^{\int P_1\,dy}=\int\left(Q_1e^{\int P_1\,dy}\right)dy+C\)
\(x\,e^{\int P_1\,dx}=\int\left(Q_1e^{\int P_1\,dx}\right)dx+C\)
The general solution of the differential equation \(e^x\,dy+(ye^x+2x)\,dx=0\) is
\(xe^y+x^2=C\)
\(xe^y+y^2=C\)
\(ye^x+x^2=C\)
\(ye^y+x^2=C\)