Differential Equations Class 12

$\frac{d^4 y}{d x^4}+\sin \left(y^{\prime \prime \prime}\right)=0$

$y^{\prime}+5 y=0$

$\left(\frac{d s}{d t}\right)^4+3 s \frac{d^2 s}{d t^2}=0$

$\left(\frac{d^2 y}{d x^2}\right)^2+\cos \left(\frac{d y}{d x}\right)=0$

$\frac{d^2 y}{d x^2}=\cos 3 x+\sin 3 x$

$\left(y^{\prime \prime \prime}\right)^2+\left(y^{\prime \prime}\right)^3+\left(y^{\prime}\right)^4+y^5=0$

$y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0$

$y^{\prime}+y=e^x$

$y^{\prime \prime}+\left(y^{\prime}\right)^2+2 y=0$

$y^{\prime \prime}+2 y^{\prime}+\sin y=0$

The degree of the differential equation
$$
\left(\frac{d^2 y}{d x^2}\right)^3+\left(\frac{d y}{d x}\right)^2+\sin \left(\frac{d y}{d x}\right)+1=0 \text { is }
$$
(A) 3
(B) 2
(C) 1
(D) not defined

The order of the differential equation
$$
2 x^2 \frac{d^2 y}{d x^2}-3 \frac{d y}{d x}+y=0 \text { is }
$$
(A) 2
(B) 1
(C) 0
(D) not defined

& $y=e^x+1$ & : & $y^{\prime \prime}-y^{\prime}=0$

& $y=x^2+2 x+\mathrm{C}$ & : & $y^{\prime}-2 x-2=0$

& $y=\cos x+\mathrm{C}$ & : & $y^{\prime}+\sin x=0$

& $y=\sqrt{1+x^2}$ & : & $y^{\prime}=\frac{x y}{1+x^2}$

& $y=\mathrm{A} x$ & : & $x y^{\prime}=y(x \neq 0)$

& $y=x \sin x$ & : & $x y^{\prime}=y+x \sqrt{x^2-y^2}(x \neq 0$ and $x>y$ or $x<-y)$

& $x y=\log y+\mathrm{C}$ & : & $y^{\prime}=\frac{y^2}{1-x y}(x y \neq 1)$

& $y-\cos y=x$ & : & $(y \sin y+\cos y+x) y^{\prime}=y$

& $x+y=\tan ^{-1} y$ & : & $y^2 y^{\prime}+y^2+1=0$

& $y=\sqrt{a^2-x^2} x \in(-a, a):$ & & $x+y \frac{d y}{d x}=0(y \neq 0)$

The number of arbitrary constants in the general solution of a differential equation of fourth order are:
(A) 0
(B) 2
(C) 3
(D) 4

The number of arbitrary constants in the particular solution of a differential equation of third order are:
(A) 3
(B) 2
(C) 1
(D) 0

$\frac{d y}{d x}=\frac{1-\cos x}{1+\cos x}$

$\frac{d y}{d x}=\sqrt{4-y^2}(-2<y<2)$

$\frac{d y}{d x}+y=1(y \neq 1)$

$\sec ^2 x \tan y d x+\sec ^2 y \tan x d y=0$

$\left(e^x+e^{-x}\right) d y-\left(e^x-e^{-x}\right) d x=0$

$\frac{d y}{d x}=\left(1+x^2\right)\left(1+y^2\right)$

$y \log y d x-x d y=0$

$x^5 \frac{d y}{d x}=-y^5$

$\frac{d y}{d x}=\sin ^{-1} x$

$e^x \tan y d x+\left(1-e^x\right) \sec ^2 y d y=0$

$\left(x^3+x^2+x+1\right) \frac{d y}{d x}=2 x^2+x ; y=1$ when $x=0$

$x\left(x^2-1\right) \frac{d y}{d x}=1 ; y=0$ when $x=2$

$\cos \left(\frac{d y}{d x}\right)=a(a \in \mathbf{R}) ; y=1$ when $x=0$

$\frac{d y}{d x}=y \tan x ; y=1$ when $x=0$

Find the equation of a curve passing through the point $(0,0)$ and whose differential equation is $y^{\prime}=e^x \sin x$.

For the differential equation $x y \frac{d y}{d x}=(x+2)(y+2)$, find the solution curve passing through the point (1, -1).

Find the equation of a curve passing through the point $(0,-2)$ given that at any point $(x, y)$ on the curve, the product of the slope of its tangent and $y$ coordinate of the point is equal to the $x$ coordinate of the point.

At any point $(x, y)$ of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (-4, -3). Find the equation of the curve given that it passes through (-2, 1).

The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after $t$ seconds.

In a bank, principal increases continuously at the rate of $r \%$ per year. Find the value of $r$ if Rs 100 double itself in 10 years $\left(\log _e 2=0.6931\right)$.

In a bank, principal increases continuously at the rate of 5\% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years $\left(e^{0.5}=1.648\right)$.

In a culture, the bacteria count is 1,00,000. The number is increased by 10\% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

The general solution of the differential equation $\frac{d y}{d x}=e^{x+y}$ is
(A) $e^x+e^{-y}=\mathrm{C}$
(B) $e^x+e^y=\mathrm{C}$
(C) $e^{-x}+e^y=\mathrm{C}$
(D) $e^{-x}+e^{-y}=\mathrm{C}$

$\left(x^2+x y\right) d y=\left(x^2+y^2\right) d x$

$y^{\prime}=\frac{x+y}{x}$

$(x-y) d y-(x+y) d x=0$

$\left(x^2-y^2\right) d x+2 x y d y=0$

$x^2 \frac{d y}{d x}=x^2-2 y^2+x y$

$x d y-y d x=\sqrt{x^2+y^2} d x$

$\left\{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)\right\} y d x=\left\{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)\right\} x d y$

$x \frac{d y}{d x}-y+x \sin \left(\frac{y}{x}\right)=0$

$y d x+x \log \left(\frac{y}{x}\right) d y-2 x d y=0$

$\left(1+e^{\frac{x}{y}}\right) d x+e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) d y=0$

$(x+y) d y+(x-y) d x=0 ; y=1$ when $x=1$

$x^2 d y+\left(x y+y^2\right) d x=0 ; y=1$ when $x=1$

$\left[x \sin ^2\left(\frac{y}{x}\right)-y\right] d x+x d y=0 ; y=\frac{\pi}{4}$ when $x=1$

$\frac{d y}{d x}-\frac{y}{x}+\operatorname{cosec}\left(\frac{y}{x}\right)=0 ; y=0$ when $x=1$

$2 x y+y^2-2 x^2 \frac{d y}{d x}=0 ; y=2$ when $x=1$

A homogeneous differential equation of the from $\frac{d x}{d y}=h\left(\frac{x}{y}\right)$ can be solved by making the substitution.
(A) $y=v x$
(B) $v=y x$
(C) $x=v y$
(D) $x=v$

Which of the following is a homogeneous differential equation?
(A) (4 x+6 y+5) d y-(3 y+2 x+4) d x=0
(B) (x y) d x-\left(x^3+y^3\right) d y=0
(C) (x^3+2 y^2\right) d x+2 x y d y=0
(D) y^2 d x+\left(x^2-x y-y^2\right) d y=0

$\frac{d y}{d x}+2 y=\sin x$

$\frac{d y}{d x}+3 y=e^{-2 x}$

$\frac{d y}{d x}+\frac{y}{x}=x^2$

$\frac{d y}{d x}+(\sec x) y=\tan x\left(0 \leq x<\frac{\pi}{2}\right)$

$\cos ^2 x \frac{d y}{d x}+y=\tan x\left(0 \leq x<\frac{\pi}{2}\right)$

$x \frac{d y}{d x}+2 y=x^2 \log x$

$x \log x \frac{d y}{d x}+y=\frac{2}{x} \log x$

$\left(1+x^2\right) d y+2 x y d x=\cot x d x(x \neq 0)$

$x \frac{d y}{d x}+y-x+x y \cot x=0(x \neq 0)$

$(x+y) \frac{d y}{d x}=1$

$y d x+\left(x-y^2\right) d y=0$

$\left(x+3 y^2\right) \frac{d y}{d x}=y(y>0)$.

$\frac{d y}{d x}+2 y \tan x=\sin x ; y=0$ when $x=\frac{\pi}{3}$

$\left(1+x^2\right) \frac{d y}{d x}+2 x y=\frac{1}{1+x^2} ; y=0$ when $x=1$

$\frac{d y}{d x}-3 y \cot x=\sin 2 x ; y=2$ when $x=\frac{\pi}{2}$

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point $(x, y)$ is equal to the sum of the coordinates of the point.

Find the equation of a curve passing through the point $(0,2)$ given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

The Integrating Factor of the differential equation $x \frac{d y}{d x}-y=2 x^2$ is
(A) $e^{-x}$
(B) $e^{-y}$
(C) $\frac{1}{x}$
(D) $x$

The Integrating Factor of the differential equation $\left(1-y^2\right) \frac{d x}{d y}+y x=a y(-1<y<1)$ is
(A) $\frac{1}{y^2-1}$
(B) $\frac{1}{\sqrt{y^2-1}}$
(C) $\frac{1}{1-y^2}$
(D) $\frac{1}{\sqrt{1-y^2}}$

Find the order and degree, if defined, of each of the following differential equations:
(i)
$$
\frac{dy}{dx}-\cos x=0.
$$

Find the order and degree, if defined, of each of the following differential equations:
(ii)
$$
xy\frac{d^2y}{dx^2}
+x\left(\frac{dy}{dx}\right)^2
-y\frac{dy}{dx}=0.
$$

Find the order and degree, if defined, of each of the following differential equations:
(iii)
$$
y'''+y^2+e^{y'}=0.
$$

Verify that the function
$$
y=e^{-3x}
$$
is a solution of the differential equation
$$
\frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0.
$$

Verify that the function
$$
y=a\cos x+b\sin x,
$$
where $a,b\in\mathbb{R}$, is a solution of the differential equation
$$
\frac{d^2y}{dx^2}+y=0.
$$

Find the general solution of the differential equation
$$
\frac{dy}{dx}=\frac{x+1}{2-y},
\qquad y\neq 2.
$$

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

Find the particular solution of the differential equation
$$
\frac{dy}{dx}=-4xy^2,
$$
given that
$$
y=1\quad\text{when}\quad x=0.
$$

Find the equation of the curve passing through the point $(1,1)$ whose differential equation is
$$
x\,dy=(2x^2+1)\,dx,
\qquad x\neq 0.
$$

Find the equation of a curve passing through the point $(-2,3)$, given that the slope of the tangent to the curve at any point $(x,y)$ is
$$
\frac{2x}{y^2}.
$$

In a bank, principal increases continuously at the rate of $5\%$ per year. In how many years will Rs 1000 double itself?

Show that the differential equation
$$
(x-y)\frac{dy}{dx}=x+2y
$$
is homogeneous and solve it.

Show that the differential equation
$$
x\cos\left(\frac{y}{x}\right)\frac{dy}{dx}
=
y\cos\left(\frac{y}{x}\right)+x
$$
is homogeneous and solve it.

Show that the differential equation
$$
2ye^{\frac{x}{y}}\,dx
+
\left(y-2xe^{\frac{x}{y}}\right)\,dy
=0
$$
is homogeneous and find its particular solution, given that
$$
x=0\quad\text{when}\quad y=1.
$$

Show that the family of curves for which the slope of the tangent at any point $(x,y)$ on it is
$$
\frac{x^2+y^2}{2xy}
$$
is given by
$$
x^2-y^2=cx.
$$

Find the general solution of the differential equation
$$
\frac{dy}{dx}-y=\cos x.
$$

Find the general solution of the differential equation
$$
x\frac{dy}{dx}+2y=x^2,
\qquad x\neq 0.
$$

Find the general solution of the differential equation
$$
y\,dx-(x+2y^2)\,dy=0.
$$

Find the particular solution of the differential equation
$$
\frac{dy}{dx}+y\cot x
=
2x+x^2\cot x,
\qquad x\neq 0,
$$
given that
$$
y=0\quad\text{when}\quad x=\frac{\pi}{2}.
$$

Find the equation of a curve passing through the point (0,1) if the slope of the tangent to the curve at any point (x,y) is equal to the sum of the x-coordinate (abscissa) and the product of the x-coordinate and y-coordinate (ordinate) of that point.

Verify that the function
$$
y=c_1e^{ax}\cos bx+c_2e^{ax}\sin bx,
$$
where $c_1,c_2$ are arbitrary constants, is a solution of the differential equation
$$
\frac{d^2y}{dx^2}
-2a\frac{dy}{dx}
+(a^2+b^2)y=0.
$$

Find the particular solution of the differential equation
$$
\log\left(\frac{dy}{dx}\right)=3x+4y,
$$
given that
$$
y=0\quad\text{when}\quad x=0.
$$

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

Solve the differential equation
$$
(\tan^{-1}y-x)\,dy
=
(1+y^2)\,dx.
$$

For each of the differential equations given below, indicate its order and degree (if defined).
(i) $\frac{d^2 y}{d x^2}+5 x\left(\frac{d y}{d x}\right)^2-6 y=\log x$

For each of the differential equations given below, indicate its order and degree (if defined).
(ii) $\left(\frac{d y}{d x}\right)^3-4\left(\frac{d y}{d x}\right)^2+7 y=\sin x$

For each of the differential equations given below, indicate its order and degree (if defined).
(iii) $\frac{d^4 y}{d x^4}-\sin \left(\frac{d^3 y}{d x^3}\right)=0$

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(i) $x y=a e^x+b e^{-x}+x^2 \quad: x \frac{}{d x^2}+2 \frac{d}{d x}-x y+x^2-2=0$

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(ii) $y=e^x(a \cos x+b \sin x): \frac{d^2 y}{d x^2}-2 \frac{d y}{d x}+2 y=0$

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(iii) $y=x \sin 3 x \quad: \frac{}{d x^2}+9 y-6 \cos 3 x=0$

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(iv) $x^2=2 y^2 \log y \quad:\left(x^2+y^2\right) \frac{d y}{d x}-x y=0$

Prove that $x^2-y^2=c\left(x^2+y^2\right)^2$ is the general solution of differential equation $\left(x^3-3 x y^2\right) d x=\left(y^3-3 x^2 y\right) d y$, where $c$ is a parameter.

Find the general solution of the differential equation $\frac{d y}{d x}+\sqrt{\frac{1-y^2}{1-x^2}}=0$.

Show that the general solution of the differential equation $\frac{d y}{d x}+\frac{y^2+y+1}{x^2+x+1}=0$ is given by $(x+y+1)=\mathrm{A}(1-x-y-2 x y)$, where A is parameter.

Find the equation of the curve passing through the point $\left(0, \frac{\pi}{4}\right)$ whose differential equation is $\sin x \cos y d x+\cos x \sin y d y=0$.

Find the particular solution of the differential equation $\left(1+e^{2 x}\right) d y+\left(1+y^2\right) e^x d x=0$, given that $y=1$ when $x=0$.

Solve the differential equation $y e^{\frac{x}{y}} d x=\left(x e^{\frac{x}{y}}+y^2\right) d y(y \neq 0)$.

Find a particular solution of the differential equation $(x-y)(d x+d y)=d x-d y$, given that $y=-1$, when $x=0$. (Hint: put $x-y=t$ )

Solve the differential equation $\left[\frac{e^{-2 \sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right] \frac{d x}{d y}=1(x \neq 0)$.

Find a particular solution of the differential equation $\frac{d y}{d x}+y \cot x=4 x \operatorname{cosec} x$ $(x \neq 0)$, given that $y=0$ when $x=\frac{\pi}{2}$.

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

The general solution of the differential equation $\frac{y d x-x d y}{y}=0$ is
(A) $x y=\mathrm{C}$
(B) $x=\mathrm{C} y^2$
(C) $y=\mathrm{C} x$
(D) $y=\mathrm{C} x^2$

The general solution of a differential equation of the type $\frac{d x}{d y}+\mathrm{P}_1 x=\mathrm{Q}_1$ is
(A) $\quad y e^{\int \mathrm{P}_1 d y}=\int\left(\mathrm{Q}_1 e^{\int \mathrm{P}_1 d y}\right) d y+\mathrm{C}$
(B) $\quad y . e^{\int \mathrm{P}_1 d x}=\int\left(\mathrm{Q}_1 e^{\int \mathrm{P}_1 d x}\right) d x+\mathrm{C}$
(C) $x e^{\int \mathrm{P}_1 d y}=\int\left(\mathrm{Q}_1 e^{\int \mathrm{P}_1 d y}\right) d y+\mathrm{C}$
(D) $x e^{\int \mathrm{P}_1 d x}=\int\left(\mathrm{Q}_1 e^{\int \mathrm{P}_1 d x}\right) d x+\mathrm{C}$

The general solution of the differential equation $e^x d y+\left(y e^x+2 x\right) d x=0$ is
(A) $x e^y+x^2=\mathrm{C}$
(B) $x e^y+y^2=\mathrm{C}$
(C) $y e^x+x^2=\mathrm{C}$
(D) $y e^y+x^2=\mathrm{C}$