Table of Contents

Exact differential equation
Homogenous linear differential equation

Exact and homogenous differential equation

Darshan M.
January 17, 2023

Table of Contents

Exact differential equation
Homogenous linear differential equation

Exact differential equation

If the differential equation of the type

$\frac{d y}{d x}=f(x, y)$

can be written in the form of

$M(x, y) d x+N(x, y) d y=0$

and

$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

then this form is called exact. The following examples clarify the procedure to solve:

For instance,

$2 x y^{3} d x+3 x^{2} y^{2} d y=0$

$M(x, y)=2 x y^{3}, \quad N(x, y)=3 x^{2} y^{2} \Rightarrow \frac{\partial M}{\partial y}=6 x y^{2}, \quad \frac{\partial N}{\partial x}=6 x y^{2}$

Now, We must find a function

$F(x, y)$

such that

$\begin{array}{l}\frac{\partial F}{\partial x}=M(x, y)=2 x y^{3}, \\\frac{\partial F}{\partial y}=N(x, y)=3 x^{2} y^{2}\end{array}$

Then, the solution would be

$F(x, y)=C$

, where C is an arbitrary constant

Homogenous linear differential equation

It involves only derivatives of y and terms involving y, and they’re set to 0, as in this equation:

$a_{n} y^{(n)}+a_{n-1} y^{(n-1)}+\ldots+a_{2} y^{\prime \prime}+a_{1} y^{\prime}+a_{0} y=0$

This equation is then converted into a characteristic equation:

$a_{n} r^{n}+a_{n-1} r^{n-1}+\ldots+a_{2} r^{2}+a_{1} r+a_{0}=0$

Depending on the roots (value of r) of this equation, we find the solution.

We would consider the case of a 2^nd order homogenous linear equation.

Case1: If both the roots are real and distinct

$k_{1} , k_{2}$

, then solution is:

$y(x)=C_{1} e^{k_{1} x}+C_{2} e^{k_{2} x}$

where

$C_{1} , C_{2}$

are arbitrary real numbers.

Case2: If both the roots are real and equal

$k_{1}$

, then solution is:

$y(x)=\left(C_{1} x+C_{2}\right) e^{k_{1} x}$

Case3: If both the roots are complex

$k_{1}=\alpha+\beta i, k_{2}=\alpha-\beta i$

, then solution is:

$y(x)=e^{\alpha x}\left[C_{1} \cos (\beta x)+C_{2} \sin (\beta x)\right]$

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About Colin Phillips

About Colin Phillips

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