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 [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are solutions to the equation [latex]y''+p(t)y'+q(t)y=0[/latex] then the linear combination [latex]y=c_{1}y_{1}+c_{2}y_{2}[/latex] is also a solution, for all constants [latex]c_{1}[/latex] and [latex]c_{2}[/latex]. If we are given initial conditions, [latex]y(a)=b[/latex] and [latex]y'(a)=b'[/latex], then we can use the initial condition to solve for [latex]c_{1}[/latex] and [latex]c_{2}[/latex] and obtain [latex]y[/latex].
Computation:
In order to solve for [latex]c_{1}[/latex] and [latex]c_{2}[/latex] and find the [latex]y[/latex], we need to have the denominator to be not zero. Hence we must have [latex]y_{1}(a)y'_{2}(a)-y_{1}'(a)y_{2}(a)[/latex] is not zero, i.e.,
[latex]\left[\begin{array}{cc} y_{1}(a) & y_{2}(a)\\ y_{1}'(a) & y_{2}'(a) \end{array}\right]\neq0.[/latex]
Definition: [latex]W=\left[\begin{array}{cc} y_{1}(t) & y_{2}(t)\\ y_{1}'(t) & y_{2}'(t) \end{array}\right][/latex] is called the Wronskian Determinate of [latex]y_{1}[/latex] and [latex]y_{2}[/latex].
Theorem: Suppose [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are solutions of the ODE, [latex]y''+p(t)y'+q(t)y=0[/latex] with initial conditions [latex]y(a)=b,y'(a)=b'[/latex]. Then it is always possible to find [latex]c_{1}[/latex] and [latex]c_{2}[/latex] such that [latex]y=c_{1}y_{1}+c_{2}y_{2}[/latex] satisfies the IVP if and only if the Wronskian [latex]W=y_{1}y_{2}'-y_{1}'y_{2}[/latex] is not zero at [latex]t=a[/latex].
Example 1: Find the Wronskian of [latex]y_{1}=e^{2t},[/latex] [latex]y_{2}=e^{-3t}[/latex].
Exercise 1: Find the Wronskian of [latex]y_{1}=e^{-t}[/latex], [latex]y_{2}=e^{4t}[/latex].
Example 2: Find the Wronskian of [latex]y_{1}=e^{2t}\text{cos}(5t),[/latex] [latex]y_{2}=e^{2t}\text{sin}(5t)[/latex].
Exercise 2: Find the Wronskian of [latex]y_{1}=e^{-t}\text{sin}(2t)[/latex], [latex]y_{2}=e^{-t}\text{cos}(2t)[/latex].
Example 3: If the Wronskian of [latex]f[/latex] and [latex]g[/latex] is [latex]2e^{2t}[/latex] and if [latex]f(t)=e^{-t}[/latex] find [latex]g(t)[/latex].
Exercise 3: If the Wronskian of [latex]f[/latex] and [latex]g[/latex] is [latex]5e^{-2t}[/latex] and if [latex]f(t)=e^{3t}[/latex] find [latex]g(t)[/latex].
Group Work
1. If the Wronskian of [latex]f[/latex] and [latex]g[/latex] is [latex]t\text{cos}(t)+\text{sin}(t)[/latex], and if [latex]u=f-3g[/latex] and [latex]v=f+g[/latex], find the Wronskian of [latex]u[/latex] and [latex]v[/latex].
2. If the Wronskian of [latex]f[/latex] and [latex]g[/latex] is [latex]2e^{3t}[/latex] and if [latex]f(t)=e^{2t}[/latex] find [latex]g(t)[/latex].
3. If the Wronskian of [latex]f[/latex] and [latex]g[/latex] is [latex]e^{t}\text{cos}(t)+\text{sin}(t)[/latex], and if [latex]u=2f-g[/latex] and [latex]v=f+2g[/latex], find the Wronskian of [latex]u[/latex] and [latex]v[/latex].
4. Find the Wronskian of [latex]y_{1}=e^{4t}\text{sin}(3t)[/latex], [latex]y_{2}=e^{4t}\text{cos}(3t)[/latex].