differential-equations

Important Points

• Order - it is the order of the highest order differential coeffecient occuring in the differential equation
• Degree - it is the power of the highest derivative occuring in the differential equation, after it has been simplified and expressed as a polynomial in derivatives i.e the powers of all the derivatives are whole numbers

  IMPORTANT

  if the degree of any derivative in the differential equation is not a whole number or any derivative is present in a function that is irremovable from the differential equation, then the degree of that differential equation will be undefined

Formation of Differential Equations

Given an equation of a family of some curves, with some arbiterary constants, we can create the differential equation of that family of curves by differentiating the given equation untill we can successfully remove all the arbiterary constants.

• The number of times we have to differentiate the given equation is equal to the number of total arbiterary independant constants present in the given equation

  order of differential equation so formed will be equal to the number of independant arbiterary constants present in the equation

• If there are 2 constants with binary operations occuring on them, then they are considered to be a single constant, $c_1+c_2=d_1$ and $c_1\times c_2=d_1$
• While removing the arbiterary constants, it is generally preferred to seprate them, so that they automatically get removed while differentiating

Solutions of Differential Equations

Given a differential equation, in order to find the family of curves that satisfy the given differential equation we can utilize integration.

• The number of times we have to integrate is equal to the order of the given differential equation
  • General Solution - a solution of the differential equation that represents the family of curves that satisfy the given differential equation, it contains arbiterary constants
  • Particular Solution - a solution of the differential equation that represents a specific curve that satisfies the given differential equation, it does not contain arbiterary constants

Variable Seprable Form

In this type of differential equation, the terms can be expressed in the following manner and solved relatively easily by just integrating both sides $f(x)dx=f(y)dy$

Reducible to Variable Seprable Form

• If a differential equation is of the following form, then it becomes variable seprable on substituting ax + by + c = t $\frac{dy}{dx}=f(ax+by+c)$
• Any differential equation of the following form can be reduced to variable seprable by substituting xy = t $yf(xy)dx+xg(xy)dy=0$

Homogenous Differential Equation

• $f(x,y)$ is said to be a homogeneous expression of degree n in the variable x and y if: $f(tx,ty)=t^{n}f(x,y)$

  • Below is a homogeneous expression of degree 2 $f(x,y)=x^2-2xy$
  • Below is a homogeneous expression of degree 3 $f(x,y)=3x^{2}y+y^3-x{y}^2$
  • Below is a homogeneous expression of degree 0 $f(x,y)=\frac{y^2}{x^2}+\sin (\frac{y}{x})$

• A differential equation of the following form (where f and g are homogeneous expressions of the same degree) is called a homogeneous differential equation $\frac{dy}{dx}=\frac{f(x,y)}{g(x,y)}$

  • Just devide the numerator and denominator with x(or y) to the power equal to the degree of both the homogeneous expressions and then just put y = tx or x = ty accordingly

Reducible To Homogenous Differential Equation

$\frac{dy}{dx}=\frac{x+y+a}{x+y+b}$ A differential equation of above format can be solved using following methods

• First Method
  • Put x = X+h and y = Y+k such that the equation becomes $\frac{dY}{dX}=\frac{X+Y+h+k+a}{X+Y+h+k+b}$
  • now h and k, doing the following will convert it into a homogeneous equation $h+k+a=0\text{and}h+k+b=0$
    • at the end plug x and y back into the equation
• Second Method (Smaller,Easier)
  • In the equation, just replace x with X and y with Y and don’t write constants such that the equation becomes a homogeneous $\frac{dY}{dY}=\frac{X+Y}{X+Y}$
  • Relation between smaller variables and bigger variables will be $X=x-h\text{and}Y=y-k$
  • Replace x with h and y with k in the original equation to get the relation between h and k to find their values $h+k+a=0\text{and}h+k+b=0$

Using Polar Coordinates for Differential Equations

$x=r\cos \theta \text{and}y=r\sin \theta$ If we do the above replacement in a differential equation then the following things become true $x^2+y^2=r^2$ $\frac{y}{x}=\tan \theta$ $xdx+ydy=rdr$ $xdy-ydx=r^{2}d\theta$

Linear Differential Equations

IMPORTANT

A differential equation is said to be linear if the dependant variable and it’s derivative occur in degree 1 only and are not multiplied together

$\frac{dy}{dx}=Py=Q\text{P,Q: functions of x}$ The solution of above equation is given by $y\times I_f=\int Q\times I_f\times dx+C$ $I_f=e^{\int Pdx}$

Reducible to Linear Differential Equation

$f^{\prime }(y)\frac{dy}{dx}+Pf(y)=Q$ We just substitute t = f(y) and it gets converted into a linear differential equation

Bernoulli’s Equation

$\frac{dy}{dx}+Py=Q{y}^n$ Just devide both sides by $y^n$ and it gets converted into a “Reducible to Linear Differential Equation”

Exact Form

$xdy+ydx=d(xy)$ $\frac{xdy+ydx}{xy}=d(\ln xy)$ $\frac{xdx+ydy}{x^2+y^2}=\frac{1}{2}d(\ln (x^2+y^2))$ $\frac{xdy-ydx}{x^2}=d(\frac{y}{x})$ $\frac{ydx-xdy}{y^2}=d(\frac{x}{y})$ $\frac{ydx-xdy}{x^2+y^2}=d({\tan }^{-1}(\frac{x}{y}))$