The differential equation $x\frac{dy}{dx}+\frac{3}{{\displaystyle \frac{dy}{dx}}}=y^2$

1. Basics of Differential Equations:

(i) Order of differential equation:

The order of a differential equation is the order of the highest differential coefficient appearing in it.

(ii) Degree of differential equation:

The degree of a differential equation is the degree of the derivative of the highest order appearing in it, after it has been expressed in a form free from radicals and fractions (as far as derivatives are concerned).

2. General and Particular Solution of Differential Equation:

A solution of a differential equation is an expression for the dependent variable in terms of the independent variable which satisfies a given relation.

The general solution includes all possible solutions and typically includes arbitrary constants or arbitrary functions.

A solution without arbitrary constants is called a particular solution.

3. Formation of Differential Equation:

If an equation in dependent and independent variables having some arbitrary constant is given, a differential equation can be obtained by differentiating the given equation w.r.t. the independent variable as many times as the number of arbitrary constants in it or by eliminating the arbitrary constants.

4. Linear Differential Equation:

The solution of linear differential equation in the form dydx+Py=Q, where P and Q are constants or functions of the independent variable x only will be y⋅e∫Pdx=∫Q⋅e∫Pdx+C, where $C$ is a constant.

The solution of linear differential equation in the form dxdy+Px=Q, where P and Q are the functions of y only will be x⋅e∫Pdy=∫Q⋅e∫Pdy+C, where $C$ is a constant.

5. Variable Separable Form:

If the differential equation can be written as $f\left(x\right)dx+g\left(y\right)dy=0$, then such differential equations are called variable-separable type. A general solution of this is given by $\int f\left(x\right)dx+\int g\left(y\right)dy=c$, where c is an arbitrary constant.

6. Homogeneous Differential Equation:

A differential equation of the form $\frac{dy}{dx}=\frac{f\left(x,y\right)}{g\left(x,y\right)},$ where $f\left(x,y\right) \& g\left(x,y\right)$ are homogeneous functions of $x$ and $y,$ i.e., of the same degree, is called homogeneous differential equation. This equation can be reduced to the form $\frac{dy}{dx}=h\left(\frac{x}{y}\right)$ & is solved by substituting y=vx, so that the dependent variable y is changed to another variable v, where v is some unknown function. Then, the differential equation is transformed to an equation with variables separable.

7. Exact Differential Equation:

Exact differential equations can be solved using the following

(i) $xdy+ydx=d\left(xy\right)$

(ii) $\frac{xdy-ydx}{x^2}=d\left(\frac{y}{x}\right)$

(iii) ydx-xdyy2=dxy

(iv) $\frac{xdy+ydx}{xy}=d\left(ln\left(xy\right)\right)$

(v) ydx-xdyxy=dlnxy

(vi) xdy-ydxxy=dlnyx

(vii) dx+dyx+y=dlnx+y

(viii) $\frac{xdy-ydx}{x^2+y^2}=d\left({\tan }^{-1}\frac{y}{x}\right)$

(ix) $\frac{ydx-xdy}{x^2+y^2}=d\left({\tan }^{-1}\frac{x}{y}\right)$

(x) xdx+ydyx2+y2=dlnx2+y2

8. Application of Differential Equations:

Orthogonal Trajectory:

Any curve that cuts every member of a given family of curves at right angles is called the orthogonal trajectory of the family.

The differential equation of the orthogonal trajectories is of the form Fx,y,-1y'=0.

The application of differential equation also lies in the problems of growth-decay, dilution, circuits, etc.