Solutions to differential equations

All $n$-th order ODEs have general solutions that depend on $n$ arbitrary constants, or parameters, such that:

$$y(t)=c_1{y}_1(t)+c_2{y}_2(t)+\cdots +c_n{y}_n(t)$$

Solutions to higher-order differential equations must satisfy the differential equation, can be expressed in terms of a general solution with arbitrary constants, and should satisfy initial problems (for an $n$-th order ODE, we have $n$ IVPs). In theory, we should be able to get all information from an IVP.

The general linear $n$-th order ODE is:

$$a_n(x)\frac{d^{n}y}{d{x}^n}+a_{n-1}(x)\frac{d^{m-1}y}{d{x}^{n-1}}+\cdots +a_1(x)\frac{dy}{dx}+a_0(x)y=g(x)$$

In general, for $a_0(x),\dots ,a_n(x)$ varying with $x$, there are no formulas for the solutions.

Fundamental set

If the $n$ solutions $y$ of the $n$-th order ODE are linearly independent, there exists a fundamental set. There must exist a fundamental set of solutions for a homogeneous linear $n$-th order ODE on an interval $I$, and it forms a basis.

By theorem, if $y$ are all solutions of the ODE, they are linearly independent if and only if the Wronskian determinant is non-zero.

Terminology

• If a solution contains $y_c=c{e}^{-x}$, it’s a transient term since $y_c\to 0$ as $x\to \infty$.
• The steady-state term is the part of the response that remains long-term.
• A singular solution is a valid solution that is lost when we solve the ODE during separation of variables.

Existence and uniqueness of solutions

We consider two fundamental problems for the solutions of differential equations: sometimes there may not be a curve for the IVP (existence) and there may be multiple curves that satisfy the IVP (uniqueness).

Continuity theorem

By theorem, if $f(t,y)$ and $\frac{\partial f}{\partial y}$ are both continuous in the general neighbourhood of $(t_0,y_0)$, then there is a unique solution for the IVP. Formally, there is some $h>0$ such that the given IVP has a unique solution for $t-t_0<t<t_0+h$.

If either of these fails to be continuous for the given initial condition, there may still be a unique solution for that IVP (i.e., they are sufficient but not necessary conditions).

For an $n$-th order ODE, $a_n(t)x^n+a_{n-1}(t)x^{n-1}+\cdots +a_1(t)x^{\prime }+a_0(t)x=g(t)$, if $a_n(t)$ and so on are continuous on an interval $I$, and $a_n(x)\ne 0$ for every $x\in I$. If $x=x_0$ (from the IVP), then a solution for the IVP exists on the interval and is unique.

Note that the following could be different: the domain of the ODE, the interval that the solution is defined or exists, and the interval of existence and uniqueness.

Interval of definition

The interval of definition is the largest interval containing $x_0$ over which the solution is defined and differentiable.

See also

• Homogeneous ODE (characteristic equation)
• Non-homogeneous ODE
  • Method of undetermined coefficients
• Euler’s method, a means of numerically approximating ODE solutions