We will talk about a type of ODE called the 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
Another important type is the 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: $$y(x)=e^{\alpha x}\left[C_{1} \cos (\beta x)+C_{2} \sin (\beta x)\right]$$
