Differential equations practice · Easy

The solution of $\left(x-y \cos \frac{y}{x}\right) d x+x \cos \frac{y}{x} d y=0$, $y(1)=0$ is given by

The solution of the differential equation $x\frac{{dy}}{{dx}} + 2y = {x^2}(x \ne 0)$ with $y(1) = 1,$ is

The solution of the differential equation $\frac{d y}{d x}+y=\cos x$ is

The solution of the differential equation $\frac{d^2 y}{d x^2}=-\frac{1}{x^2}$ is

Solution of the differential equation $y\left(x y+2 x^2 y^2\right) d x+x\left(x y-x^2 y^2\right) d y=0$ is given by

Let $y = y(x)$ be the solution of the differential equation, $\frac{{dy}}{{dx}} + y\tan x = 2x + {x^2}\tan x,x \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$, such that y(0)=1. Then

The solution of the differential equation $\cos x \sin y \,d x+\sin x \cos y \,d y=0$, is

The solution of $\log \left( {\frac{{dy}}{{dx}}} \right) = ax + by$ is

$y=c_1 e^x-c_2 e^{-x}$ is the general solution of a differential equation. The order of the differential equation is , where $c_1$ and $c_2$ are arbitrary constants.

If ${c_1},{c_2},{c_3},{c_4},{c_5}$ and ${c_6}$ are constants, then the order of the differential equation whose general solution is given by
$y = {c_1}\cos (x + {c_2}) + {c_3}\sin (x + {c_4}) + {c_5}{e^x} + {c_6}$ is

If $x^2+y^2=1$, then $\left[\right.$ Note $\left.: y^{\prime}=\frac{\mathrm{dy}}{\mathrm{dx}}, \mathrm{y}^{\prime \prime}=\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}\right]$

Let the solution curve $y = y\left( x \right)$ of the differential equation
$\left[ {\frac{x}{{\sqrt {{x^2} - {y^2}} }} + {e^{\frac{y}{x}}}} \right]x\frac{{dy}}{{dx}} = x + \left[ {\frac{x}{{\sqrt {{x^2} - {y^2}} }} + {e^{\frac{y}{x}}}} \right]y$
pass through the points (1, 0) and $\left( {2\alpha ,\alpha } \right),\,\,\alpha > 0.$ Then $\alpha$ is equal to

If $\frac{d y}{d x}=f(x)+\int_0^1 f(x) d x$, then the equation of the curve $y=f(x)$ passing through $(0,1)$ is

The elimination of the arbitrary constant $m$ from the equation $y=e^{m x}$ gives the differential equation

The degree of the differential equation $y_3^{2/3} + 2 + 3{y_2} + {y_1} = 0$, is

General solutions of $y^{\prime}=e^{y / x}+y / x$ is

Which of the following statements on ordinary differential equations is/are true?
(i) The number of arbitrary constants is same as the degree of the differential equation.
(ii) A linear differential equation can contain products of the dependent variable and its derivatives.
(iii) A particular integral cannot contain arbitrary constants.
(iv) By putting $v=\frac{y}{x}$ any homogeneous first order differential equation transforms to variable separable form.

The differential equation of the family of lines through the origin, is

Which of the following transformation reduce the differential equation $\frac{\mathrm{d} z}{\mathrm{~d} x}+\frac{z}{x} \log z=\frac{z}{x^2}(\log z)^2$ into the form $\frac{\mathrm{d} u}{\mathrm{~d} x}+u P(x)=Q(x)$

The differential equation of the family of non-vertical lines is .......

If $y = a\sin (5x + c)$, then

Let $y=y(x)$ be the solution of the differential equation $x^{3} d y+(x y-1) d x=0, x>0, y\left(\frac{1}{2}\right)=3-e$. Then $y(1)$ is equal to

A solution of $y=x y^{\prime}-3 y^{\prime 3}$ is

If the curve $y = y(x)$ is the solution of the differential equation
$2({x^2} + {x^{5/4}})dy - y(x + {x^{1/4}})\mathrm{ }dx = 2{x^{9/4}}dx,\,\,x > 0$ which passes through the point $\left( {1,\,\,1 - \frac{4}{3}{{\log }_e}2} \right)$,then the value of $y(16)$ is equal to

The value of the constant ' $m$ ' and ' $c$ ' for which $y=m x+c$ is a solution of the differential equation $D^2 y-3 D y-4 y=-4 x$.

If $y=ce^{\sin ^{-1} x}$, then corresponding to this the differential equation is

When $x=0, y=1$ the particular solution of the differential equation $\left(1+y^2\right) d x+\left(1+x^2\right) d y=0$ is

Statement 1: The degrees of the differential equations $\frac{d y}{d x}+y^2=x$ and $\frac{d^2 y}{d x^2}+y=\sin x$ are equal.

Statement 2: The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest positive integral power of the highest order derivative involved in the differential equation, otherwise degree is not defined.

The differential equation of the family of parabolas having vertex at the origin and the $y$-axis as the axis of symmetry, is

The general solution of $y^{\prime}+\sin \frac{x+y}{2}=\sin \frac{x-y}{2}$ is