Differential Equations Overview

Differential Equations

An equation involving differentials, differential coefficients, or derivatives is called a differential equation.
Examples of differential equations include:
- dydx=x+3 \frac{dy}{dx} = x + 3 dxdy=x+3
- d2ydx2+3dydx+2y=0 \frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0 dx2d2y+3dxdy+2y=0
- y+xydzdy=k(1+dydx)2 y + xy\frac{dz}{dy} = k\left(1 + \frac{dy}{dx}\right)^2 y+xydydz=k(1+dxdy)2
- p=d2y/dx21+dy/dx p= \frac{d^2y/dx^2}{1+dy/dx}p=1+dy/dxd2y/dx2
- ydydx=xdydx+a y \frac{dy}{dx} = x\frac{dy}{dx} + a ydxdy=xdxdy+a
Differential equations refer to those equations where all differential coefficients have reference to a single independent variable.
Partial differential equations contain differential coefficients related to more than one independent variable, which are not discussed here.

Order and Degree of Differential Equations

The order of a differential equation is determined by the order of the highest derivative appearing in it.
Example: The order of the equation 1+(dydx)3/2=pd2ydx2 1 + (\frac{dy}{dx})^{3/2} = p\frac{d^2 y}{dx^2}1+(dxdy)3/2=pdx2d2y is two because the highest derivative d2ydx2 \frac{d^2 y}{dx^2} dx2d2y is present.
The degree of a differential equation corresponds to the degree of the highest derivative in the equation after simplifying it by removing radicals and fractions.

Solutions of Differential Equations

A solution or integral of a differential equation represents a relationship between the dependent and independent variables, where the derivatives obtained from the relationship satisfy the given equation.
Example: y=sinx+cy = sin x + cy=sinx+c is a solution of the differential equation dydx=cosx \frac{dy}{dx} = cos xdxdy=cosx.

Kinds of Solutions

General Solution or the Complete Primitive: The solution of a differential equation containing a number of arbitrary constants equal to the order of the differential equation is called the general solution or the complete primitive.
Example: y=c1cosx+c2sinxy = c_{1} cos x + c_{2} sinxy=c1cosx+c2sinx is a general solution of the differential equation d2ydx2+y=0 \frac{d^2y}{dx^2} + y = 0 dx2d2y+y=0
Particular Solution: If particular values are assigned to the arbitrary constants in the general solution, the resulting solution is called the particular solution.
Example: For the complete primitive y=Acos(x+B) y = A cos (x + B)y=Acos(x+B), when constants AAA and BBB are assigned the specific values 111 and 000, the particular solution is y=cosxy = cos xy=cosx
Singular Solution: Solutions that cannot be derived from the general solution, even by assigning specific values to the arbitrary constants, are called Singular Solutions.

Examples

1.* Find the differential equation of the family of curves y=Ae2x+Be−2xy = Ae^{2x} + Be^{-2x}y=Ae2x+Be−2x, for different values of A and B.
Solution:
- Start with the equation: y=Ae2x+Be−2xy = Ae^{2x} + Be^{-2x}y=Ae2x+Be−2x
- Differentiate both sides with respect to xxx: dydx=2Ae2x−2Be−2x \frac{dy}{dx} = 2Ae^{2x}-2Be^{-2x}dxdy=2Ae2x−2Be−2x
- Differentiate again: d2ydx2=4Ae2x+4Be−2x=4(Ae2x+Be−2x)=4y \frac{d^2y}{dx^2} = 4Ae^{2x} + 4Be^{-2x} = 4(Ae^{2x} + Be^{-2x}) = 4ydx2d2y=4Ae2x+4Be−2x=4(Ae2x+Be−2x)=4y.
- Therefore, the required differential equation is d2ydx2=4y \frac{d^2y}{dx^2} = 4y dx2d2y=4y
2.* Find the differential equation of the system of curves y=ax2+bcosnx+c y=ax^2 + b cos nx + cy=ax2+bcosnx+c, where a, b, c are arbitrary constants.
Solution:
- We have y=ax2+bcosnx+c y=ax^2+b cos nx+cy=ax2+bcosnx+c.
- Differentiate: dydx=2ax−bnsinnx \frac{dy}{dx} = 2ax - bn sin nxdxdy=2ax−bnsinnx
- Differentiate again: d2ydx2=2a−bn2cosnx \frac{d^2y}{dx^2} = 2a - bn^2 cos nxdx2d2y=2a−bn2cosnx
- Differentiate once more: d3ydx3=bn3sinnx \frac{d^3y}{dx^3} = bn^3 sin nxdx3d3y=bn3sinnx
- Eliminate 'a' between equations (ii) and (iii): x2yd2ydx2−dydx=bn(cosnx−sinnx) x^{2}y\frac{d^2y}{dx^2} - \frac{dy}{dx} = bn(cos nx - sin nx)x2ydx2d2y−dxdy=bn(cosnx−sinnx)
- Eliminate 'b' between equations (iv) and (v) by dividing:
  d3ydx3/x2yd2ydx2−dydxx2y \frac{d^3y}{dx^3} / \frac{x^2y\frac{d^2y}{dx^2}-\frac{dy}{dx}}{x^2y}dx3d3y/x2yx2ydx2d2y−dxdy = −x2sinnxcosnx−sinnx \frac{-x^2 sin nx}{cos nx -sin nx}cosnx−sinnx−x2sinnx
- This is the required differential equation. Notice that three constants have been eliminated, resulting in a differential equation of the third order.

Exercises

Find the differential equation from x2−y2+2λxy=1x^2 - y² + 2\lambda xy = 1x2−y2+2λxy=1, where λ is a parameter.
Determine a differential equation from the equation ax2+by2=1ax^2 + by^2 = 1ax2+by2=1, where a and b are parameters.
Find the differential equation of all circles of radius r.
Form the differential equation of all circles of radius r.
Show that v = A/r + B is a solution of d2vdr2 \frac{d^2v}{dr^2}dr2d2v + 2rdvdr=0 \frac{2}{r}\frac{dv}{dr} = 0r2drdv=0.
Find the differential equation corresponding to y = ae^2x + be^-3x + ce^x.

Differential Equations of First Order and First Degree

2.1 Equations of the First Order and First Degree

The general form of an ordinary differential equation of the first order and first degree can be written as M+Ndydx=0M + N \frac{dy}{dx} = 0M+Ndxdy=0, where M and N are functions of x and y, or are constants.
It can also be expressed as:
- dydx=f(x,y)\frac{dy}{dx} = f(x, y)dxdy=f(x,y)
- (x,y)dy=f(x,y)dx(x, y) dy = f(x, y) dx(x,y)dy=f(x,y)dx
- −f(x,y)dx+(x,y)dy=0-f(x, y) dx + (x, y) dy = 0−f(x,y)dx+(x,y)dy=0
- $M...