14.10: Differential equations


Differential Equations

When an equation is produced with differentials in it it is called a differential equation. The final force equation produced for parachute person based of physics is a differential equation.

Integral and Integro-differential equation

If an equation instead has integrals then it is an integral equation and if an equation has both derivatives and integrals it is known as an integro-differential equation. Integral equations and integro-differential equations can be converted into differential equations to be solved or alternatively you can use Laplace equations to solve the equations.

Difference equation

Many differential equations are solvable analytically however when the complexity of a system increases it is usually an intractable problem to solve differential equations and this leads us to using numerical methods. In this case the differential equations reduce down to a difference equation. We used numerical methods for parachute person but we did not need to in that particular case as it is easily solvable analytically, it was more of an academic exercise.

From parachute person let us review the differential equation and the difference equation that was generated from basic physics.

 $$m \frac{dv}{dt} = mg - c v^2$$ $$v_{i+1} = v_i + (g - \frac{c}{m}(v_i)^2)(t_{i+1}-t_i)$$ Differential equation from parachute person. This is a first order differential equation. Difference equation from parachute person.

Integro-differential equation and RLC circuit

To complete this initial discussion we look at electrical engineering and the ubiquitous RLC circuit is defined by an integro-differential equation if we use Kirchhoff's voltage law. This can be converted to a differential equation as show in the table below.

 $$-Ri(t)-L\frac{di(t)}{dt}-\frac{1}{C}\int_{-\infty}^t i(t')dt'+V(t)=0$$ $$RC\frac{dv_c(t)}{dt}+LC\frac{d^2v_c(t)}{dt}+v_c(t)=V(t)$$ Assume a current of i(t) produced with a voltage V(t) we get this integro-differential equation for a serial RLC circuit. The equation to the left is converted into a differential equation by specifying the current in the capacitor as $$C\frac{dv_c(t)}{dt}$$ where $$v_c(t)$$ is the voltage across the capacitor. The current in the capacitor would be dthe current for the whole circuit.

First-order differential equation

First order systems are divided into natural response and forced response parts.

• Natural response is called a homogeneous solution or sometimes a complementary solution, however we believe the natural response name gives a more physical connection to the idea.
• Force response is called a particular solution in mathematics. Again force response as more of a physical connection.

Let us take an simple first-order differential equation as an example.

$\tau\frac{dx(t)}{dt}+x(t)=K_s f(t)$

We solve this problem in two parts, the natural response part and then the force response part.

• Natural response part

$\tau\frac{dx(t)}{dt}+x(t)=0$

$\frac{dx_n(t)}{dt}=-\frac{x_n(t)}{\tau}$

Let's rewrite this in order to integrate.

$\frac{dx_n(t)}{x_n(t)}=-\frac{dt}{\tau}$

$\int \frac{dx_n(t)}{x_n(t)}=-\int \frac{dt}{\tau}$

$x_n(t)=e^{-\frac{t}{\tau}} e^c$

$x_n(t)=\alpha e^{-\frac{t}{\tau}}$

• Force response part
• For a general function f(t) there would be many solutions so...
• We will assume a constant force

$\tau\frac{dx_f(t)}{dt}+x_f(t)=K_s F$

The solution to this is obvious as the derivative of a constant is zero so we just set $$x_f(t)$$ to $$K_s F$$. Therefore $$x_f(t)=K_s F$$ for $$t \ge 0$$.

• Full solution

$x(t) = x_n(t)+x_f(t)=\alpha e^{-\frac{t}{\tau}} + K_s F$

Force equation idea versus mathematical idea

This is a defense of the idea of using natural and force response as opposed to the more mathematical definitions (which is appropriate in a pure math course, but this is engineering/science class).

A homogeneous differential equation of order n is

$f_n(x)y^{(n)}+f_{n-1}(x)y^{n-1} \ldots f_1(x)y'+f_0(x)y=0 where $$y^{n}$$ is the $$n_{th}$$ derivative of the function y. A non-homogeneous differential equation of order n is \[f_n(x)y^{(n)}+f_{n-1}(x)y^{n-1} \ldots f_1(x)y'+f_0(x)y=g(x)$

The solution to the non-homogeneous equation is

$y(x)=y_c(x)+y_p(x)$ where $$y_c(x)$$ is the complementary solution of the homogenous differential equation and where $$y_p(x)$$ is the particular solutions based off g(x).

The term complementary is for the solution and clearly means that it complements the full solution.

We, however, like to take a physical interpretation and call the complementary solution a natural solution and the particular solution a forced solution. However it should be noted that this is contrary to mathematical definitions (natural means something else in mathematics). Under this terminology the solution to the non-homogeneous equation is

$y(x)=y_n(x)+y_f(x)$ where $$y_n(x)$$ is the natural (or unforced) solution of the homogenous differential equation and where $$y_f(x)$$ is the forced solutions based off g(x).

The idea for these terms comes from the idea of a force equation for a spring-mass-damper system.

$m\ddot{x} + B\ddot{x} + kx = K_s F(x)$

where m is mass, B is the damping coefficient, and k is the spring constant and $$m\ddot{x}$$ is the mass force, $$B\ddot{x}$$ is the damper force, and $$kx$$ is the spring force (Hooke's law).

Natural solution, complementary solution, and homogeneous solution to a homogeneous differential equation are all equally valid. Forced solution and particular solution are as well equally valid. There is no need for a debate, just some understanding that there are different definitions.

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