Introduction to Differential Equations

What is a differential equation?
An equation that involves one or more derivatives of an unknown function is called a differential equation. The order of the highest derivative included in a differential equation defines the order of this equation.

Examples and Detailed Solutions

Example 1
a) $$y' = 3x$$,
the order of the highest derivative is 1 ($$y'$$), so the order of this differential equation is 1.
b) $$y'' + y' + y = 3x$$,
the order of the highest derivative is 2 ($$y''$$), so the order of this differential equation is 2.
c) $$-2y''' + y'' + y^4 = 3x$$,
the order of the highest derivative is 3 ($$y'''$$), so the order of this differential equation is 3.

$$y = f(x)$$ is a solution of a differential equation if the equation is satisfied upon substitution of $$y$$ and its derivatives into the differential equation.

Example 2
Verify that $$y = Ce^{4x} + e^{3x}$$, where $$C$$ is a constant, is a solution to the differential equation
$y' - 4y = -e^{3x}$
$$y'$$ is given by
$$y' = 4Ce^{4x} + 3e^{3x}$$
We now substitute $$y'$$ and $$y$$ into the left side of the equation and simplify
$$y' - 4y = 4Ce^{4x} + 3e^{3x} - 4(Ce^{4x} + e^{3x})$$
$$= 4Ce^{4x} + 3e^{3x} - 4Ce^{4x} - 4e^{3x}$$
$$= 4Ce^{4x} - 4Ce^{4x} + e^{3x}(3 - 4)$$
$$= -e^{3x}$$
Which is equal to the left side of the given equation and therefore $$y = Ce^{4x} + e^{3x}$$ is a solution to the differential equation $$y' - 4y = -e^{3x}$$.

Most of the work on differential equations consists in solving these equations. For example, to solve the following differential example
$y' = 2x$
Let us integrate both sides of the given equation as follows
$\int y' \, dx = \int 2x \, dx$
which gives
$$y + C_1 = x^2 + C_2$$
where $$C_1$$ and $$C_2$$ are constants of integration. The solution $$y$$ of the above equation is given by: $$y = x^2 + C$$, where $$C = C_2 - C_1$$.

References

Differential Equations
Differential Equations - Runge Kutta Method