Section 2.2 Wronskian of a differential equation

Objective:

1. The Theorem of super position and IVP.

2. Definition of Wronskian of a differential equation

Recall that if $y_1$ and $y_2$ are solutions to the equation $y^{″}+p(t)y^{\prime }+q(t)y=0$ then the linear combination $y=c_1{y}_1+c_2{y}_2$ is also a solution, for all constants $c_1$ and $c_2$. If we are given initial conditions, $y(a)=b$ and $y^{\prime }(a)=b^{\prime }$, then we can use the initial condition to solve for $c_1$ and $c_2$ and obtain $y$. 

Computation:

In order to solve for $c_1$ and $c_2$ and find the $y$, we need to have the denominator to be not zero. Hence we must have $y_1(a)y_2^{\prime }(a)-y_1^{\prime }(a)y_2(a)$ is not zero, i.e., 

$\left[\begin{array}{cc}{y}_1(a)& y_2(a)\\ y_1^{\prime }(a)& y_2^{\prime }(a)\end{array}\right]\ne 0.$

Definition: $W=\left[\begin{array}{cc}{y}_1(t)& y_2(t)\\ y_1^{\prime }(t)& y_2^{\prime }(t)\end{array}\right]$ is called the Wronskian Determinate  of $y_1$ and $y_2$. 

Theorem: Suppose $y_1$ and $y_2$ are solutions of the ODE, $y^{″}+p(t)y^{\prime }+q(t)y=0$ with initial conditions $y(a)=b,y^{\prime }(a)=b^{\prime }$. Then it is always possible to find $c_1$ and $c_2$ such that $y=c_1{y}_1+c_2{y}_2$ satisfies the IVP if and only if the Wronskian $W=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2$ is not zero at $t=a$. 

Example 1: Find the Wronskian of $y_1=e^{2t},$ $y_2=e^{-3t}$.

Exercise 1: Find the Wronskian of $y_1=e^{-t}$, $y_2=e^{4t}$. 

Example 2: Find the Wronskian of $y_1=e^{2t}\text{cos}(5t),$ $y_2=e^{2t}\text{sin}(5t)$.

Exercise 2: Find the Wronskian of $y_1=e^{-t}\text{sin}(2t)$, $y_2=e^{-t}\text{cos}(2t)$. 

Example 3: If the Wronskian of $f$ and $g$ is $2e^{2t}$ and if $f(t)=e^{-t}$ find $g(t)$.

Exercise 3: If the Wronskian of $f$ and $g$ is $5e^{-2t}$ and if $f(t)=e^{3t}$ find $g(t)$.

Group Work

1. If the Wronskian of $f$ and $g$ is $t\text{cos}(t)+\text{sin}(t)$, and if $u=f-3g$ and $v=f+g$, find the Wronskian of $u$ and $v$.

2. If the Wronskian of $f$ and $g$ is $2e^{3t}$ and if $f(t)=e^{2t}$ find $g(t)$.

3. If the Wronskian of $f$ and $g$ is $e^t\text{cos}(t)+\text{sin}(t)$, and if $u=2f-g$ and $v=f+2g$, find the Wronskian of $u$ and $v$.

4. Find the Wronskian of $y_1=e^{4t}\text{sin}(3t)$, $y_2=e^{4t}\text{cos}(3t)$.