Differential equations practice · Mixed difficulty

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}, x \in\left(0, \frac{\pi}{2}\right)$ satisfying the condition $y\left(\frac{\pi}{4}\right)=2$. Then, $y\left(\frac{\pi}{3}\right)$ is

Solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=\sin (x+y)+\cos (x+y) \text { is }$

The solution of the differential equation $\frac{{dy}}{{dx}} = \frac{1}{{{x^2} + {y^2}}}$ is

The solution of $\frac{{dy}}{{dx}} + \sqrt {\left( {\frac{{1 - {y^2}}}{{1 - {x^2}}}} \right)} = 0$ is

The solution of $y^{\prime}=\frac{2 x-y+1}{x-2 y+1}$ is given by

A particles moves in a straight line with a velocity given by $\frac{{dx}}{{dt}} = x + 1$ ($x$ is the distance described). The time taken by a particle to traverse a distance of 99 metres is

Assertion : The differential equation

$\frac{d x}{d y}+x=\cos y \text { and } \frac{d x}{d y}+\frac{-2 x}{y}=y^2 e^{-y}$

are first order linear differential equations.

Reason : The differential equation of the form

$\frac{{dx}}{{dy}}+{P}_1 {x}={Q}_1$

where, $P_1$ and $Q_1$ are constants or functions of $y$ only, is called first order linear differential equation.

The general solution of $x \frac{d y}{d x}+(\log x) y=x^{1-\frac{1}{2} \log x}$ is

The solution of $\frac{d y}{d x}+y=e^{-x}, y(0)=0$, is

The differential equation of family of circles with fixed radius 5 units and centre on the line $y=2$, is

The degree of the differential equation
$y(x)=1+\frac{d y}{d x}+\frac{1}{1.2}\left(\frac{d y}{d x}\right)^2+\frac{1}{1.2 .3}\left(\frac{d y}{d x}\right)^3+\ldots \ldots$. Is

The order and degree of the differential equation $\frac{{{d^2}y}}{{d{x^2}}} = \sqrt[3]{{1 - {{\left( {\frac{{dy}}{{dx}}} \right)}^4}}}$ are respectively

On putting $\frac{y}{x}=v$ the differential equation $\frac{d y}{d x}=\frac{2 x y-y^2}{2 x y-x^2}$ is transformed to

The solution to the differential equation $(x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^2$ is

The solution of the differential equation $\frac{{dy}}{{dx}} - \frac{{y + 3x}}{{{{\log }_e}\left( {y + 3x} \right)}} + 3 = 0$ is (where $C$ is a constant of integration)

Consider the differential equation $e^x(y d x-d y)=e^{-x}(y d x+d y)$. Let ${y}={f}({x})$ be the particular solution of the differential equation which passes through $(0,2)$. Then range of $g(x)=\log _2(f(x))$, is

The solution of the differential equation $y \mathrm{dx}-\left(x+2 y^2\right) \mathrm{dy}$ $=0$ is $x=\mathrm{f}(y)$. If $\mathrm{f}(-1)=1$, then $\mathrm{f}(\mathrm{a})$ is equal to :

The differential equation of all parabolas whose axes are parallel to axis of $x$, is

The solution of the differential equation $\frac{{dy}}{{dx}} = \frac{{x + y}}{x}$ satisfying the condition $y\left( 1 \right) = 1$ is

The differential equation which represents the family of curves $y=C_1 e^{C_2 x}$, where $C_1$ and $C_2$ are arbitrary constants, is

The solution of the differential equation $y d x-x d y+3 x^2 y^2 e^{x^3} d x=0$ is

To solve first order linear differential equation, we use following steps

I. Write the solution of the given differential equation as

$y(I F)=\int(\mathrm{Q} \times \text { IF }) d x+C$

II. Write the given differential equation in the from $\frac{d y}{d x}+P y=\mathrm{Q}$, where $P$ and $Q$ are constants or functions of $x$ only.

III. Find the integrating factor (IF) $\mathrm{e}^{\int \mathrm{P} \mathrm{dx}}$

The correct order of the above steps is

The equation of one of the curves whose slope at any point is equal to $y + 2x$ is

Let $F$ denotes the family of ellipses whose centre is at the origin and major axis is the $y$-axis. Then, equation of the family $F$ is

The differential equation for the family of curves ${x^2} + {y^2} - 2ay = 0$, where $a$ is an arbitrary constant, is

Let $y = y\left( x \right)$ be the solution of the differential equation $\sin x\frac{{dy}}{{dx}} + y\cos x = 4x,x \in \left( {0,\pi } \right)$
If $y\left( {\frac{\pi }{2}} \right) = 0$, then $y\left( {\frac{\pi }{6}} \right)$ is equal to

The degree of the differential equation satisfied by the curve $\sqrt{1+\mathrm{x}}-\mathrm{a} \sqrt{1+\mathrm{y}}=1$, is

The order of the differential equation whose solution is $y=a \cos x+b \sin x+c e^{-x}$ is

Solution of the equation $\frac{d y}{d x}=\sqrt{y-x}$ is

The general solution of ${y^2}dx + \left( {{x^2} - xy + {y^2}} \right)dy = 0$ is