3.2 Solution of Linear Homogeneous Equations; the Wronskian

Differential Operator

Let p and q be continuous functions on an open interval I , for $\alpha < t < \beta$. The cases for $\alpha = -\infty$or $\beta = \infty$or both , are included.Then for any function $\phi$ that is twice differential on I , we define the differential operator L by the equation

$$L[\phi] = \phi'' + p \phi' + q \phi$$

for example ,if $p(t) = t^2 ,\ q(t) = 1+t,\ \phi(t) = sin3t$

$$L[\phi](t) = (sin3t)'' + t^2(sin3t)' + (1+t)sin3t$$

we usually write this equation in the form

$$L[y] = y''+p(t)y'+q(t)y = 0$$

with the initial value $y(t_0) = y_0,\quad y'(t_0) = y_0'$

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Theorem 3.2.1 Existence and uniqueness Theorem

Consider the initial value problem

$$y'' + p(t) y' + q(t) y = g(t),\quad y(t_0) = y_0, \quad y'(t_0) = y_0'$$

where p, q ,and g are continuous on an open interval I that contains the point $t_0$. This problem has exactly one solution $y = \phi (t)$,and the solution exists throughout the interval I.

• Note : the initial problem has an unique solution on an interval I

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Theorem 3.2.2 Principle of Superposition

if $y_1,\ y_2$are two solutions of the differential equation

$$L[y] = y'' + p(t) y' + q(t)y = 0$$

then the linear combination $c_1y_1 + c_2 y_2$is also a solution for any value of the constants $c_1 ,\ c_2$

proof

$$\begin{split} L[c_1y_1 + c_2 y_2] &= [c_1y_1 + c_2 y_2]'' + p[c_1y_1 + c_2 y_2]' + q[c_1y_1 + c_2 y_2]\\ &= c_1y_1'' + c_2y_2'' + c_1py_2'+ c_2py_2'+ c_1qy_1 + c_2q y_2\\ &=c_1[y_1'' + py_1' + q y_1] + c_2[y_2'' + py_2' + q y_2]\\ &=c_1L[y_1] + c_2L[y_2] \end{split}$$

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Note

• $W[y_1 ,y_2](x)$ is not everywhere zero iff $c_1 y_1 + c_2 y_2$ contains all solutions of equation.
• Therefore, $y = c_1 y_1(t) + c_2 y_2(t)$is called general solution
• $y_1$and $y_2$ are said to form a fundamental set of solutions iff $Wy_1 ,y_2 \neq 0$