# Differential equations

Analytical solutions to first-order linear ODEs.

Differential equations contain functions as well as their derivatives. The order of the equation corresponds to the order of the highest derivative.

• An ordinary differential equation (ODE) contains functions of only one independent variable.
• A partial differential equation (PDE) contains functions of more than one independent variables.

# First-order linear ODEs

Standard form: $$y'+p(t)\,y=q(t)$$

Where $$t$$ is the independent variable (in models this is often time).

The equation is homogeneous if $$q(t)=0$$ , and inhomogeneous otherwise.

## Homogeneous first-order linear ODEs

Can always be solved by separation of variables:

$$\cfrac{dy}{dt}+p(t)\,y=0$$ becomes $$\cfrac{dy}{y}=-p(t)dt$$

General solution: $$y=c\,e^{-P(t)}$$

Steps: Integrating both sides gives $$\int\cfrac{1}{y}dy=-\int p(t)dt$$ , which is $$ln|y|=-P(t) + C$$ with $$P(t)=\int p(t)dt$$ . Then solve for $$y$$ :

$$|y|=e^{-P(t) + C}$$ $$y=\pm e^{-P(t) + C}$$ $$y=\pm e^{-P(t)}e^{C}$$ $$y=c\,e^{-P(t)}$$

## Inhomogeneous first-order linear ODEs

Can be solved by variation of parameters.

General solution: $$y=u(t)\,y_h$$

Where $$y_h$$ is a non-zero solution to the associated homogeneous ODE, and $$u$$ is determined as follows. Substitute $$y=u\,y_h$$ into the inhomogeneous equation:

$$\cfrac{d}{dt}(uy_h) + puy_h=q$$

Applying the product rule:

$$u'y_h+uy_h'+puy_h=q$$ $$u'y_h+u(y_h'+py_h)=q$$

Note that $$y_h'+py_h=0$$ , because $$y_h$$ is a solution to the homogeneous ODE. Thus $$u'y_h=q$$ . Then integrate $$u'=\cfrac{q}{y_h}$$ , in order to solve for $$u$$ .

This is equivalent to finding the integrating factor. The idea is to find a factor $$u$$ such that when multiplying both sides by it ( $$uy'+puy=qu$$ ), the left-hand side corresponds to $$(uy)'$$ after applying the product rule. This is the case when $$u'=pu$$ , which is a homogeneous ODE that can be solved by separation of variables and gives as integrating factor:

$$u=e^{\int p(t)dt}$$

After calculating the integrating factor $$u$$ , multiply both sides of the equation by $$u$$ , and rewrite the left-hand side as $$(uy)'$$ . Then integrate.

## Linear combinations

A linear combination of functions $$f_1,f_2,...$$ are functions of the form $$c_1f_1 + c_2f_2 + ...$$ with $$c_1,c_2,...$$ any numbers.

Superposition principles:

That is, for linear homogeneous ODEs, all linear combinations of solutions are solutions:

• 0 is a solution.
• Multiplying any solution by a scalar gives another solution.
• Adding any two solutions gives another solution.

Also the general inhomogeneous solution $$y=y_p+y_h$$ , where

• $$y_p$$ is a particular inhomogenous solution and
• $$y_h$$ is the general homogeneous solution.

## Special cases

If we have an inhomogeneous linear ODE where $$q(t)$$ is constant: If there is a constant solution, then $$y'=0$$ and the equation simplifies to $$p(t)y=q$$ , thus one particular solution is $$y=\frac{q}{p(t)}$$ . Using the superposition principle, all solutions take the form:

$$y=\frac{q}{p(t)} + Ce^{p(t)t}$$

Solutions to $$z^n=1$$ with $$z$$ any complex number are $$e^{i(\frac{2\pi k}{n})}$$ for $$k=0,1,...n-1$$ .

Solutions to $$x''+x=0$$ are all linear combinations of sine and cosine: $$x=c_1\text{cos}(t)+c_2\text{sin}(t)$$ with $$c_1,c_2$$ any real number.