Ncert Solutions for Class 12
Differential Equations
Differential Equations
1. For each of the differential equations given below, indicate its order and degree (if defined).
(i) (frac{d^2y}{dx^2}+5xleft(frac{dy}{dx}right)^2-6y=log{x})
(ii) (left(frac{dy}{dx}right)^3-4left(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 : xfrac{d^2y}{dx^2}+2frac{dy}{dx}-xy+x^2-2=0)
(ii) (y=e^xleft(acos{x}+bsin{x}right) : frac{d^2y}{dx^2}-2frac{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^2right)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^2right)dy left(yneq0right)).
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^{-2sqrt x}}{sqrt x}-frac{y}{sqrt x}right]frac{dx}{dy}=1 left(xneq0right)).
11. Find a particular solution of the differential equation (frac{dy}{dx}+ycot{x}=4xcos{e}cx left(xneq0right)),
given that y=0 when (x=frac{pi}{2}).
12. Find a particular solution of the differential equation (left(x+1right)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+2xright)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)