Differential equations practice · Easy

The differential equation $\frac{{dy}}{{dx}} + Py = Q{y^n}$ , $n > 2$ can be reduced to linear form by substituting

Solution of the equation $\frac{d y}{d x}+\frac{y(x+y)}{x^2}=0$ is -

Let $y=[a \cos x+(b+c) \sin x] e^{x+d}$ where $a, b, c, d$ are parameters. The order of the differential equation formed by eliminating of parameters is

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

The general solution of the differential equation $\frac{d y}{d x}=e^{y+x}+e^{y-x}$ is

If $x^4+y^4=x^2 y^2$ then $\frac{d y}{d x}$ is given by

The solution of $y^{\prime}(y-x-4)=x+y-2$ is given by

The solution to the differntial equation $y d x+x \log \left(\frac{y}{x}\right) d y-2 x d y=0$ is

Assertion : The degree of the differential equation

$\frac{d^3 y}{d x^3}+2\left(\frac{d^2 y}{d x^2}\right)^{\frac{3}{2}}+2 y=0 \text { is zero }$

Reason: The degree of a differential equation is not defined if it is not a polynomial eq in its derivatives.

The degree of the differential equation $\frac{d y}{d x}+x=$ $\left(y+x \frac{d y}{d x}\right)^{-4}$ is

Let $f(x)$ be differentiable on the interval $(0, \infty)$ such that $f(1)=1$, and $\lim\limits _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$. Then $f(x)$ is

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

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

Let the solution curve $y=y(x)$ of the differential equation $\frac{d y}{d x}-\frac{3 x^5 \tan ^{-1}\left(x^3\right)}{\left(1+x^6\right)^{3 / 2}} y=2 x \exp \left\{\frac{x^3-\tan ^{-1} x^3}{\sqrt{\left(1+x^6\right)}}\right\}$ pass through the origin. Then $y(1)$ is equal to

Let $y(x)$ be the solution of the differential equation $(x \ln x) \frac{d y}{d x}+y=2 x \ln x,(x \geq 1)$. Then $y(e)$ is equals to:

Consider the following statements

I. The order of the differential equation $\frac{d y}{d x}=e^x$ is 1 .

II. The order of the differential equation $\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}+\mathrm{y}=0$ is 2 .

III. The order of the differential equation

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

Choose correct option.

The integrating factor of the differential equation $(x \log x) \frac{d y}{d x}+y=2 \log x$ is

The equation of the curve whose subnormal is twice the abscissa, is

The solution of the differential equation $\left(x y^2-e / x^3\right) d x=x^2 y d y ; y(1)=0$, is -

Solution of the differential equation

$x=1+x y \frac{d y}{d x}+\frac{(x y)^2}{2 !}\left(\frac{d y}{d x}\right)^2+\frac{(x y)^3}{3 !}\left(\frac{d y}{d x}\right)^3+\ldots$
is

Statement-1: The slope of the tangent at any point $\mathrm{P}$ on a parabola, whose axis is the axis of $x$ and vertex is at the origin, is inversely proportional to the ordinate of the point $P$.
Statement-2: The system of parabolas $y^2=4 a x$ satisfies a differential equation of degree 1 and order 1 .

If $\left(1+x^2\right) \frac{d y}{d x}=x(1-y), y(0)=\frac{4}{3}$, then the value of $y(\sqrt{8})+\frac{8}{9}$ is

The solution of $x^{\prime}+x \tan t=\sec t, x(0)=1$ is

If the function $y = {\sin ^{ - 1}}x$, then $\left( {1 - {x^2}} \right)\frac{{{d^2}y}}{{d{x^2}}}$ is equal to

The differential equation representing the family of curves ${y^2} = 2c\left( {x + \sqrt c } \right)$, where $c > 0$ is a parameter is of order and degree as follows

If $(2+\sin x) \frac{d y}{d x}+(y+1) \cos x=0$ and $y(0)=1$, then $y\left(\frac{\pi}{2}\right)$ is equal to :

If $(2+\sin x) \frac{d y}{d x}+(y+1) \cos x=0$ and $y(0)=1$, then $y\left(\frac{\pi}{2}\right)$ is equal to

The solution of differential equation $(\sin x + \cos x)dy + (\cos x - \sin x)dx = 0$ is

If $c$ is any arbitrary constant, then the general solution of the differential equation $y d x-x d y=x y d x$ is given by

The integrating factor of the differential equation $\frac{{dy}}{{dx}} + y = \frac{{1 + y}}{x}$, is