Differential equations

The following notations for derivatives are used: and , that is .

Definition. A differential equation is a relation between an independent variable x, the unknown function and its derivatives , of the form

If the unknown function $y$ is a function of a single variable x, the differential equation is called ordinary.

Definition. A solution of a differential equation on an interval (a,b) is a function $y=ϕ(x)$ defined on that interval together with its derivatives, and for which substituting $y=ϕ(x)$ into the differential equation turns it into an identity in x on (a,b).

To determine all the functions which are solutions of a differential equation is to solve that differential equation.

A separable differential equation has the form: where $F:I\to \mathbb{R}$ and $g:J\to \mathbb{R}$ are two continuous functions. The equation is equivalent to $g(y)dy=f(x)dx$. Integrating, we obtain $(G\circ y)(x)=F(x)+c$ and where G is an antiderivative of g, F is an antiderivative of f, and $c\in \mathbb{R}$.

A first-order linear differential equation has the form: $y'=p(x)*y+q(x),$ where $p,q:I\to \mathbb{R}$ are continuous functions. The solution of this equation is a differentiable function $ϕ:I\to \mathbb{R}$ satisfying: $ϕ'(x)=p(x)\cdot ϕ(x)+q(x), \forall x\in I.$

First-order differential equations: , where p, t are continuous functions on an interval (a,b).

Every solution of the first-order ordinary equation is the sum of a particular solution of that equation and the general solution of the equation

is determined for constant and for a particular form of , as shown in the table below.

| | | | Form of | Form of the particular solution | | Polynomial | Polynomial of the same degree | | | | | | | | $acos(rx)+bsin(rx)$ | |

Solving the homogeneous equation:

The homogeneous equation , $p:I\to \mathbb{R}$ has the solution $ϕ:I\to \mathbb{R},$ $\forall x\in I$.

The second-order differential equation has the form $F(x,y,y',y)=0.$

Equations of the form $ay''+by'+cy=h(x)$ are called second-order differential equations with constant coefficients.

The second-order linear homogeneous differential equation

The equation with has at least one solution $f:\mathbb{R}\to \mathbb{R}$ of the form . The solutions of the characteristic equation, are , and a and b are real numbers determined from the conditions and

We have the following cases:

	solutions of the equation	solution of the equation

Graphical representation of the solutions of differential equations

1. A body heated to is immersed in a medium held at a constant temperature of . Within one minute the temperature of the body falls to . Let be the temperature of the body at time t and the constant temperature of the medium. Then satisfies the differential equation: , that is , which has the analytic solution: . From the initial conditions and .

From these conditions we obtain: . Therefore