Methods of Solving First Order, First Degree Differential Equation

Solve the following differential equation $\frac{dy}{dx}=\frac{1+x^2}{1+y^2}$

Solve the differential equation $\frac{dy}{dx}=\frac{2x}{y^2}$

Solve the differential equation $\frac{dy}{dx}=-\left[\frac{ x+y\cos x}{1+\sin x}. \right]$

Solve the differential equation $\frac{dy}{dx}=1+\frac{y}{x}$

Solve the differential equation, $\mathrm{x}(\mathrm{x}+\mathrm{y})\mathrm{dy}=({\mathrm{x}}^2+{\mathrm{y}}^2)\mathrm{dx}.$

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}+\mathrm{y}+8}{2(\mathrm{x}+\mathrm{y})+3}$

Find a particular solution of the differential equation given below

$\begin{array}{rl}& ({\tan }^{-1}\mathrm{y}-\mathrm{x})\mathrm{dy}=(1+{\mathrm{y}}^2)\mathrm{dx}\end{array}$ given that $\mathrm{y}=0$ when $\mathrm{x}=0$.

$\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}=\mathrm{y}$

Solve the differential equation

$\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}={\mathrm{x}}^2{\mathrm{y}}^4$

Find the general solution of the differential equation $x\frac{dy}{dx}+2y=x^2 \left(x\ne 0\right)$.

Solution of the differential equation $\frac{dy}{dx}=\frac{se{c}^{2}x+2x{e}^{-x}+\tan x}{2y{e}^{y^{2-x}}}$ is ?

Following are the differential equations and their respective solutions. Select the incorrect one.

$\left(\mathrm{I}\right) y\text{'}=a\sin 2x-\sin x: y=a{\sin }^{2}x+\cos x+9$

$\left(\mathrm{II}\right) y\text{'}=\frac{b}{2x\sqrt{\log x}}: y=b\sqrt{\log x}+2$

$\left(\mathrm{III}\right) y\text{'}=-7x+3{\sec }^{2}3x+5e^x\cos 5x+e^x\sin 5x: y=-7+e^x\sin 5x+\tan 3x$

If area of triangle formed by tangent (with positive slope) and normal to a curve at any point in first quadrant with $x-$axis is cube of its ordinate,then differential equation of such family is:

The general solution of the differential equation $y\text{'}=\frac{4}{x\left(x-4\right) }$ is$?$

Find the particular solution of the differential equation $\log \left(\frac{dy}{dx}\right)=3x+4y$ given that $y=0$ when $x=0$.

Find the equation of a curve passing through the point $\left(0, 1\right)$. If the slope of the tangent to the curve at any point $\left(x, y\right)$ is equal to the sum of the $x$ coordinate (abscissa) and the product of the $x$ coordinate and $y$ coordinate (ordinate) of that point.

Find the general solution of the differential equation $y dx-\left(x+2y^2\right)dy=0$.

Show that the family of curves for which the slope of the tangent at any point $\left(x, y\right)$ on it is $\frac{x^2+y^2}{2xy}$, is given by $x^2-y^2=cx$.

Show that the differential equation $2y e^{\frac{x}{y}}dx+\left(y-2x e^{\frac{x}{y}}\right)dy=0$ is homogeneous and find its particular solution, given that, $x=0$ when $y=1$.

Show that the differential equation $x \cos \left(\frac{y}{x}\right)\frac{dy}{dx}=y \cos \left(\frac{y}{x}\right)+x$ is homogeneous and solve it.