General Solutions |
Consider the linear homogeneous \(n\)th order ODE
\[y^{(n)}+p_1(x)y^{(n-1)}+p_2(x)y^{(n-2)}+\cdots+p_n(x)y=0. \;\;\;\;(43)\]
Principle of superposition: If \(y_1,y_2,\ldots,y_n\) are solutions of (43), then \(c_1y_1+c_2y_2+\cdots+c_ny_n\)
is also a solution of (43). (verify)
Theorem.
If \(y_1,y_2,\ldots,y_n\) are linearly independent solutions of (43), i.e., \(W(y_1,\,y_2,\ldots,y_n)\neq 0\),
then the general solution of (43) is
\[y=c_1y_1+c_2y_2+\cdots+c_ny_n,\]
for arbitrary constants \(c_1,\ldots,c_n\).
Any \(n\) linearly independent solutions of (43) are called fundamental solutions of (43)
because any solution of (43) is a linear combination of them. Note the general formula for Wronskian:
\[W(y_1,\,y_2,\ldots,y_n)=\,\begin{array}{|llcl|}
y_1& y_2& \cdots& y_n\\
y_1'& y_2'& \cdots& y_n'\\
\vdots& \vdots& \cdots& \vdots\\
y_1^{(n-1)}& y_2^{(n-1)}& \cdots& y_n^{(n-1)}\end{array}\,.\]
Example.
Show that \(e^x,\cos x\), and \(\sin x\) are linearly independent functions.
Solution.
\[\begin{align*}
W(e^x,\cos x,\sin x) &= \,\begin{array}{|ccc|}
e^x & \cos x & \sin x\\
(e^x)' & (\cos x)' & (\sin x)'\\
(e^x)'' & (\cos x)'' & (\sin x)''
\end{array} \\
&= \,\begin{array}{|ccc|}
e^x & \cos x & \sin x\\
e^x & -\sin x & \cos x\\
e^x & -\cos x & -\sin x
\end{array} \\
&= e^x\, \begin{array}{|cc|}
-\sin x & \cos x\\
-\cos x & -\sin x\end{array}
-\cos x\,\begin{array}{|cc|}
e^x & \cos x\\
e^x & -\sin x\end{array}
+\sin x\,\begin{array}{|ccc|}
e^x & -\sin x \\
e^x & -\cos x \end{array} \\
&= e^x(\sin^2x+\cos^2x)-\cos x(-e^x\sin x-e^x\cos x) +\sin x(-e^x\cos x+e^x\sin x) \\
&= e^x(\sin^2x+\cos^2x)+e^x(\sin^2x+\cos^2x)\\
&= 2e^x \neq 0.
\end{align*}\]
Since \(W(e^x,\cos x,\sin x)\neq 0\), \(e^x,\cos x\), and \(\sin x\) are linearly independent functions.
Consider the linear nonhomogeneous \(n\)th order ODE
\[y^{(n)}+p_1(x)y^{(n-1)}+p_2(x)y^{(n-2)}+\cdots+p_n(x)y=g(x). \;\;\;\;(44)\]
Theorem.
If \(y_1,y_2,\ldots,y_n\) are fundamental solutions of the corresponding homogeneous ODE of (44) and
\(y_p\) is a particular solution of (44), then the general solution of (44) is
\[y=c_1y_1+c_2y_2+\cdots+c_ny_n+y_p,\]
for arbitrary constants \(c_1,\ldots,c_n\).
Consider the IVP
\[\begin{align}
y^{(n)}+p_1(x)y^{(n-1)}+p_2(x)y^{(n-2)}+\cdots+p_n(x)y&=g(x) \nonumber\\
y(x_0)=a_0,y'(x_0)=a_1,\ldots,y^{(n-1)}(x_0)&=a_{n-1}. \;\;\;\;(45)
\end{align}\]
Theorem.
If \(p_1,p_2,\ldots,p_n\), and \(g\) are continuous functions on an interval containing \(x_0\),
then the ODE (45) has a unique solution \(y=\phi(x)\) on interval containing \(x_0\).
Consider the linear homogeneous \(n\)th order ODE with constant coefficients \[y^{(n)}+a_1y^{(n-1)}+a_2y^{(n-2)}+\cdots+a_ny=0. \;\;\;\;(46)\] The characteristic equation of (46) is \[r^n+a_1r^{n-1}+\cdots+a_n=0.\]
Example.
Solve the IVP
\[\begin{align*}
y'''-2y''+y'-2y&=0\\
y(0)=3,y'(0)=3,y''(0)&=7.
\end{align*}\]
Solution. The characteristic equation is
\[\begin{align*}
r^3-2r^2+r-2&=0\\
r^2(r-2)+(r-2)&=0\\
(r-2)(r^2+1)&=0\\
r&=2,\pm i.
\end{align*}\]
So the general solution is
\[y=c_1e^{2x}+c_2\cos x+c_3\sin x.\]
Taking derivatives we get
\[\begin{align*}
y'&=2c_1e^{2x}-c_2\sin x+c_3\cos x\\
y''&=4c_1e^{2x}-c_2\cos x-c_3\sin x.
\end{align*}\]
Using the initial conditions \(y(0)=3,y'(0)=3,y''(0)=7\), we get
\[\begin{align*}
c_1+c_2 &=3\\
2c_1 +c_3&=3\\
4c_1-c_2 &=7.
\end{align*}\]
Solving we get \(c_1=2, c_2=1, c_3=-1\). Thus the solution is
\[y=2e^{2x}+\cos x-\sin x.\]
Example.
Find the general solution of
\[\begin{align*}
(D-2)^3(D^2-2D+2)^2y&=0.
\end{align*}\]
Here the differential operator \(D\) is defined as \(D^ky=\displaystyle\frac{d^ky}{dx^k}\).
Solution. The characteristic equation is
\[\begin{align*}
(r-2)^3(r^2-2r+2)^2&=0\\
r&=2,2,2,1\pm i,1\pm i.
\end{align*}\]
So the general solution is \[y=c_1e^{2x}+c_2xe^{2x}+c_3x^2e^{2x}+c_4e^x\cos x+c_5e^x\sin x+c_6xe^x\cos x+c_7xe^x\sin x.\]
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