Exact and homogenous differential equation

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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 2nd 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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