Differential Equations

The order of the differential equation corresponding to the family of parabolas whose axes are along the $X$-axis and whose foci are at the origin, is

The differential equation of the family of hyperbola having their centres at origin and their axes along the coordinates axes is

The general solution of the differential equation $y y^{\prime}=x\left[\frac{y^2}{x^2}+\frac{\phi\left(\frac{y^2}{x^2}\right)}{\phi^{\prime}\left(\frac{y^2}{x^2}\right)}\right]$, where $\phi$ is an arbitrary function, is

Let $c_1, c_2, c_3, c_4$ be arbitrary constants. The order of the differential equation, corresponding to $y=c_1 e^x+$ $\mathrm{c}_2 \mathrm{e}^{\log _{\mathrm{e}} \mathrm{x}}+\mathrm{c}_3 \sin ^2…

$a, b, c, d$ are real numbers. The general solution of $\frac{d y}{d x}=\frac{a x+b}{c y+d}$ represents a family of straight lines, when

$y=Ae^{x}+Be^{-2x}$ satisfies which of the following differential equations?

The general solution of $\frac{d y}{d x}=\cos ^2(x-y-1)$ is given by $\mathrm{x}=$

If \(c_1, c_2, c_3, c_4, c_5\) are arbitrary constants, then the order of the differential equation whose general solution is \(y=\left(c_1+c_2\right) \sin \left(x+c_3\right)+c_4 e^{x+c_5}\) is

If the differential equation obtained by eliminating A,B from $y=\left(\sin ^{-1} x\right)^2+A \cos ^{-1} x+B$ is $\left(a-x^2\right) y^{\prime \prime}-x y^{\prime}=b$, then $\frac{b+a}{b-a}=$

If $\frac{a}{a_1}=\frac{b}{b_1}$, then the substitution to be used to solve the differential equation $\frac{d y}{d x}=\frac{a x+b y+c}{a_1 x+b_1 y+c_1}$ by using separation of variables is

Find the sum of the order and degree of the differential equation \(y=x\left(\frac{d y}{d x}\right)^3+\frac{d^2 y}{d x^2}\)

If $2x-y+c\log\,\left(\left(x-2y-4\right)\right)=k$is the general solution of $\frac{dy}{dx}=\frac{2x-4y-5}{x-2y+2}$then$c=$