1. 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) \(\frac{d^4y}{dx^4}-sin{\left(\frac{d^3y}{dx^3}\right)}=0\)

2. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i) \(y=ae^x+be^{-x}+x^2 : x\frac{d^2y}{dx^2}+2\frac{dy}{dx}-xy+x^2-2=0\)
(ii) \(y=e^x\left(acos{x}+bsin{x}\right) : \frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0\)
(iii) \(y=xsin{3}x : \frac{d^2y}{dx^2}+9y-6cos{3}x=0\)
(iv) \(x^2=2y^2log{y} : \left(x^2+y^2\right)\frac{dy}{dx}-xy=0\)

3. 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.

4. Find the general solution of the differential equation \(\frac{dy}{dx}+\sqrt{\frac{1-y^2}{1-x^2}}=0\).

5. 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.

6. Find the equation of the curve passing through the point \(\left(0,\frac{\pi}{4}\right)\) whose differential equation is \(sinxcosydx+cosxsinydy=0\).

7. Find the particular solution of the differential equation \((1+e^{2x})dy+(1+y^2)e^xdx=0\),
given that y=1 when x=0.

8. Solve the differential equation \(ye^{\frac{x}{y}}dx=\left(xe^{\frac{x}{y}}+y^2\right)dy \left(y\neq0\right)\).

9. Find a particular solution of the differential equation \((x–y)(dx+dy)=dx–dy\), given that y=–1, when x=0.

10. Solve the differential equation \(\left[\frac{e^{-2\sqrt x}}{\sqrt x}-\frac{y}{\sqrt x}\right]\frac{dx}{dy}=1 \left(x\neq0\right)\).

11. Find a particular solution of the differential equation \(\frac{dy}{dx}+ycot{x}=4xcos{e}cx \left(x\neq0\right)\),
given that y=0 when \(x=\frac{\pi}{2}\).

12. Find a particular solution of the differential equation \(\left(x+1\right)\frac{dy}{dx}=2e^{-y}-1\), given that y=0 when x=0.

13. The general solution of the differential equation \(\frac{ydx-xdy}{y}=0\) is
(A) \(xy=C\)                    (B) \(x=Cy^2\)                 (C) \(y=Cx\)                      (D) \(y=Cx^2\)

14. The general solution of a differential equation of the type \(\frac{dx}{dy}+P_1x=Q_1\) is
(A)  \(ye^{\int{P_1dy}}=\int{\left(Q_1e^{\int{P_1dy}}\right)dy+C}\)
(B) \(ye^{\int{P_1dx}}=\int{\left(Q_1e^{\int{P_1dx}}\right)dx+C}\)
(C) \(xe^{\int{P_1dy}}=\int{\left(Q_1e^{\int{P_1dy}}\right)dy+C}\)
(D) \(xe^{\int{P_1dx}}=\int{\left(Q_1e^{\int{P_1dx}}\right)dx+C}\)

15. The general solution of the differential equation \(e^xdy+\left(ye^x+2x\right)dx=0\)
(A) \(xe^y+x^2=C\)       (B) \(xe^y+y^2=C\)         (C) \(ye^x+x^2=C\)         (D) \(ye^y+x^2=C\)