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 $f\left(0\right)=3$

The differential equation of the family of ellipses having foci on the $y$-axis and centre at the origin is

Let $a$ and $b$ be real numbers. For which of the following differential equations is $xy=a{e}^x+b{e}^{-x}+x^2$ a solution?

Which of the following is TRUE about the differential equation

${\left(\frac{d^{3}y}{d{x}^3}\right)}^2+7{\left(\frac{d^{2}y}{d{x}^2}\right)}^3+2{\sin }^{2}y=0$?

If $f^{\text{'}}\left(x\right)=\frac{1}{x}$ then $f\left(3\right)-f\left(2\right)$ is equal to

Let $a$ and $b$ be arbitrary constants. A differential equation representing a family of curves $y=\cos \left(ax\right)+\sin \left(bx\right)$ is

A differential equation having $y=x$ as a particular solution is

The degree and the order of the differential equation

$-{\left(\frac{d^{3}y}{d{x}^3}\right)}^2+2{\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{d^{2}y}{d{x}^2}\right)}^4+y=0$

respectively are

The differential equation of the family of hyperbolas having centre at the origin and foci on the $x$-axis is

Which of the following is $\mathrm{TRUE}$concerning the order and degree of the differential equation ${\left(\frac{d^{3}y}{d{x}^3}\right)}^2+{\sin }^{2}x{\left(\frac{d^{2}y}{d{x}^2}\right)}^3+\cos x{\left(\frac{dy}{dx}\right)}^5+y^3=\tan x?$

The differential equation of the family of circles with centre on the $x$-axis is

The particular integral of the differential equation $f\left(D\right)y=e^{ax}$, where $f\left(D\right)=\left(D-a\right)g\left(D\right),g\left(a\right)\ne 0$ is

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 $y=x\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2$.

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

Differential equation of family of ellipse whose centre is $\left(1,0\right)$ and major axis is $x$-axis is

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 ...........