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

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

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

If m and n are the order and degree of the differential equation.

${\left(\frac{{\text{d}}^2\text{y}}{{\text{dx}}^2}\right)}^5+\frac{4{\left(\frac{{\text{d}}^2\text{y}}{{\text{dx}}^2}\right)}^3}{\frac{{\text{d}}^3\text{y}}{{\text{dx}}^3}}+\frac{{\text{d}}^3\text{y}}{{\text{dx}}^3}={\text{x}}^2-1$ , then :

If m and n are the order and degree of the differential equation.

${\left(\frac{{\text{d}}^2\text{y}}{{\text{dx}}^2}\right)}^5+\frac{4{\left(\frac{{\text{d}}^2\text{y}}{{\text{dx}}^2}\right)}^3}{\frac{{\text{d}}^3\text{y}}{{\text{dx}}^3}}+\frac{{\text{d}}^3\text{y}}{{\text{dx}}^3}={\text{x}}^2-1$ , then :

The order of the differential equation of all tangent lines to the parabola $\mathrm{y}={\mathrm{x}}^2$ is

A solutions of the differential equation ${\left(\frac{dy}{dx}\right)}^2-\frac{dy}{dx}\left(e^x+e^{-x}\right)+1=0$ may be

If the general solution of some differential equation is $\mathrm{y}={\mathrm{a}}_1\left({\mathrm{a}}_2+{\mathrm{a}}_3\right)·\cos \left(\mathrm{x}+{\mathrm{a}}_4\right)-{\mathrm{a}}_5{\mathrm{e}}^{\mathrm{x}+{\mathrm{a}}_6}$ then order of differential equation is ...........