TeachingDifferential equations – Ordinary differential equations – Second-order linear differential equations
A second-order linear differential equation has the form
\[P(t)y''(t)+Q(t)y'(t)+R(t)y(t)=G(t),\]where \(P\), \(Q\), \(R\) and \(G\) are continuous functions with \(P(t)\not\equiv0\). This differential equation is called homogeneous if \(G(x)\equiv0\) and nonhomogeneous if \(G(x)\not\equiv0\).
Since \(P(t)\not\equiv0\) a second-order linear differential equation can be written in the standard form
\[y''(t)+p(t)y'(t)+q(t)y(t)=g(t)\quad\text{with}\quad p(t)=\frac{Q(t)}{P(t)},\quad q(t)=\frac{R(t)}{P(t)}\quad\text{and}\quad g(t)=\frac{G(t)}{P(t)}.\]Now we only consider intervals where the functions \(p,\), \(q\) and \(g\) are continuous. Then we have:
Theorem: Consider the initial-value problem
\[y''(t)+p(t)y'(t)+q(t)y(t)=g(t),\quad y(t_0)=y_0,\quad y'(t_0)=y_0',\]where \(p\), \(q\) and \(g\) are continuous functions on an open interval \(I\) that contains the point \(t_0\). This initial-value problem has a unique solution and this solution exists throughout the interval \(I\).
This is called the existence and uniqueness theorem. This theorem says that there exists a solution (existentie), that this solution is unique (uniqueness) and that this exists throughout the interval \(I\).
Fo a homogeneous linear differential equation we have the principle of superposition:
Theorem: If \(y_1\) and \(y_2\) are solutions of \[y''(t)+p(t)y'(t)+q(t)y(t)=0,\]
then \(y(t)=c_1y_1(t)+c_2y_2(t)\) is aslo a solution for any values of \(c_1,c_2\in\mathbb{R}\).
Proof: If \(y_1\) and \(y_2\) are solutions, then we have: \(y_1''(t)+p(t)y_1'(t)+q(t)y_1(t)=0\) and \(y_2''(t)+p(t)y_2'(t)+q(t)y_2(t)=0\).
Then we have for \(y(t)=c_1y_1(t)+c_2y_2(t)\):
If \(y(t)=c_1y_1(t)+c_2y_2(t)\) satisfies the initial values \(y(t_0)=y_0\) en \(y'(t_0)=y_0'\), then we have:
\(\left\{\begin{array}{l}c_1y_1(t_0)+c_2y_2(t_0)=y_0\\[2.5mm]c_2y_1'(t_0)+c_2y_2'(t_0)=y_0'.\end{array}\right.\)
This system has a unique solution if the coeffcient matrix \(\begin{pmatrix}y_1(t_0)&y_2(t_0)\\y_1'(t_0)&y_2'(t_0)\end{pmatrix}\)
is invertible and therefore if
Definition: The determinant \(W(y_1,y_2)(t_0):=\begin{vmatrix}y_1(t_0)&y_2(t_0)\\y_1'(t_0)&y_2'(t_0)\end{vmatrix} =y_1(t_0)y_2'(t_0)-y_2(t_0)y_1'(t_0)\) is called the Wronskian determinant or the Wronskian of the solutions \(y_1\) and \(y_2\).
We conclude that the two functions \(y_1\) and \(y_2\) are linear independent if and only if \(W(y_1,y_2)(t)\neq0\).
Now we have Abel's theorem:
Theorem: If \(y_1\) and \(y_2\) are solutions of the homogeneous differential equation \[y''(t)+p(t)y'(t)+q(t)y(t)=0\]
with \(p\) and \(q\) continuous functions on an open interval \(I\), then we have for all \(t\in I\):
\[W(y_1,y_2)(t)=c\cdot e^{-\int p(t)\,dt}\quad\text{for some}\quad c\in\mathbb{R}.\]Note that this means that: \(c\neq0\) if and only if \(y_1\) and \(y_2\) are linear independent.
Proof: If \(y_1\) and \(y_2\) are solutions, then we have: \(y_1''(t)+p(t)y_1'(t)+q(t)y_1(t)=0\) and \(y_2''(t)+p(t)y_2'(t)+q(t)y_2(t)=0\).
If we multiply the second equation by \(y_1(t)\) and the first one by \(-y_2(t)\) and we add the results, then we have:
From \(W(t)=y_1(t)y_2'(t)-y_2(t)y_1'(t)\) we deduce that \(W'(t)=y_1'(t)y_2'(t)+y_1(t)y_2''(t)-y_2'(t)y_1'(t)-y_2(t)y_1''(t)=y_1(t)y_2''(t)-y_2(t)y_1''(t)\).
Hence: \(W'(t)+p(t)W(t)\). This implies that \(W(t)=c\cdot\exp\left(-\int p(t)\,dt\right)\) for some \(c\in\mathbb{R}\).
Corollary: \(W(y_1,y_2)(t)\) is either zero for all \(t\in I\) (if \(c=0\)) or never zero for all \(t\in I\) (if \(c\neq0\)).
Last modified on April 19, 2021
