Category: Basic Concepts

Introduction to Differential Equations

To put it simply, a differential equation is any equation that contains derivatives. For example, `(y'')/t+y'+ty=0` and `(y''')^4+sqrt(y'')-y'=5t` are both differential equations.

Different notations can be used: either `y^((n))` or `(d^ny)/(dt^n)`.

Existence and Uniqueness of the Solution the ODE

This note contains some theorems that refer to the existence and uniqueness of the solution to the ODE.

Theorem 1. Consider the n-th-order linear differential equation: `y^((n))+p_1(t)y^((n-1))+p_2(t)y^((n-2))+...+p_n(t)=f(t)`. If all coefficients `p_1(t)`, `p_2(t)`, ..., `p_n(t)` and `f(t)` are continuous on the interval `(a,b)`, the equation has the unique solution which satisfies the given initial conditions `y(t_0)=y_0`, `y'(t_0)=y_0^'`, ..., `y^((n-1))(t_0)=y_0^((n-1))`, where `t_0` belongs to the interval `(a,b)`.