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# Differential equations

## Definitions and examples of differential equations

An ordinary differential equation, abbreviated ODE, is an equation that involves derivatives of a function. The order of the equation is the order of the highest derivative in the equation.

ODE are used frequently to model real-world phenomena.

• The simplest ODE defines the primitive $x$ of a function $f$ $$\frac{d}{dt} x(t) = f(t)$$
• The rate of decay of radioactive atoms is proportional to their number $N(t)$ over time. It satisfies the first order ODE $$\dot{N} = -\lambda N$$ (the dot represents the derivative with respect to time).
• The force on a string is inversely proportional to its position $x(t)$ away from its equilibrium. By Newton's law, its acceleration $\ddot{x}$ is proportional to the force. This gives the second order ODE $$m \ddot{x} = - k x.$$
• The Volterra equation models the evolution of a population of prey $x(t)$ and predators $y(t)$. It is a first order ODE. $$\dot{x} = x(1-y),\quad \dot{y} = -y(1-x).$$

## Solutions of a differential equation

In general, a solution to a differential equation is not unique. For instance the solutions to the equation $$\dot{x} = -x$$ are the functions of the form $$x(t) = x_0 e^{-t}$$ for any $x_0$. Indeed, $$\dot{x}(t) = -x_0e^{-t} = -x(t).$$

• A general solution to the ODE is the collection of all the solutions expressed using a parameter. Here $x_0 e^{-t}$ is the solution, and the parameter is $x_0$.
• A particular solution is a solution for a given value of the parameter, for example $2e^{-t}$.
• The solution of a first order ODE is unique if we impose an initial condition, i.e. the value of the solution at a given point. For instance the solution to the ODE with initial condition $x(1)=1$ is $e^{1-t}$.
• The number of parameters is generally equal to order of the ODE.
Solutions to $\dot{x} = t.$ General solutions are illustrated by the family of curves. The particular solution curve going through $(0,0)$ is determined by the condition $x=0$ when $t=0$

## Techniques for solving differential equations

We detail classical techniques for solving differential equations.

• Direct integration The solutions to $$\dot{x}(t) = f(t)$$ are the primitives $\displaystyle x(t) = \int f(t)\,dt$.

The solutions to the $\ddot{x}(t) = g$ are obtained by integrating twice. This gives $x(t) = g\, t^2/2 + x_1 t + x_0$.

• Separation of variables The solutions to $$\dot{x}(t) = \frac{g(t)}{f(x)}$$ satisfy $f(x)\,dx = g(t)\,dt$. Thus, $\displaystyle \int f(x)\,dx = \int g(t)\,dt.$

The solutions to $\dot{x} = 2xt$ solve $dx / x = 2t\,dt$. We integrate both sides to find $\ln\vert x\vert = t^2 + c$, hence $x = c' e^{t^2}$.

• Reduction by substitution where we change the function to be differentiated.

To solve $\dot{x} - 2 t x = \cos t$, we set $y = e^{-t^2}x$. We get $$\dot{y} = e^{-t^2}\dot{x} -2t e^{-t^2}x = e^{-t^2}(\dot{x} - 2t x) = e^{-t^2}\cos t.$$ So, $\displaystyle y(t) = \int e^{-t^2}\cos t\,dt.$