Basics of Differential Equations

If $f\left(x\right)f^{\text{'}}\left(-x\right)=f\left(-x\right)f^{\text{'}}\left(x\right)$, then the value of $f\left(x\right)f\left(-x\right)$ for all $x$ is given $f\left(0\right)=3$

Identify the independent variables of the partial differential equation $\frac{\partial u}{\partial y}=-\frac{\partial u}{\partial x}$.

Find the singular solution of the differential equation $y=x\frac{dy}{dx}-{\left(\frac{dy}{dx}\right)}^3$.

Find the singular solution for differential equation $y=x\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2+2$.

Find the singular solution of $y=x\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2+\frac{dy}{dx}$.

Find the singular solution of $y=x\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2$.

What are the independent variables of the following Partial Differential Equation.

$x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=u$

What are the independent variables of the following Partial Differential Equation.

$\frac{{\partial }^{2}u}{\partial x^2}+{\left(\frac{{\partial }^{2}u}{\partial x\partial y}\right)}^2+\frac{{\partial }^{2}u}{\partial y^2}=x^2+y^2$

What are the independent variables of the following Partial Differential Equation.

$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$

What are the independent variables of the following Partial Differential Equation.

$\frac{{\partial }^{2}u}{\partial t^2}-c^2\frac{{\partial }^{2}u}{\partial x^2}=0$

A differential equation is considered to be partial differential equation if it has

The order and the degree of the differential equation of all ellipses with centre at the origin, major axis along $X$-axis and eccentricity $\frac{\sqrt{3}}{2}$ are, respectively

Degree of differential equation $e^{\frac{dy}{dx}}=x$ is:

The differential equation satisfied by the family of curves given by $y=ax\cos \left(\frac{1}{x}+b\right)$ is...

$($where $a, b$ are arbitrary constant$)$

The differential equation of all parabolas each of which has a latus rectum 4a and whose axes are parallel to the Y-axis is

The order and degree of the differential equation whose solution is $y=cx+c^2-3c^{\frac{3}{2}}+2,$ where $c$ is a parameter is

The differential equation of all straight lines passing through the origin is

If $\frac{dy}{dx}=y+3$ and $y\left(0\right)=2,$ then $y\left(\ln 2\right)$ is equal to

The differential equation of the family of curves $y=ax+\frac{1}{a},$where $a\ne 0$ is an arbitrary constant, has the degree

Differential equation of all family of lines $y=mx+\frac{4}{m}$ by eliminating the arbitrary constant $m$ is