Section 2.3. Fundamental solutions and Abel’s theorem

Objective:

1. Definition of a fundamental set of solutions

2. Abel’s Theorem, finding Wronskian

In previous section 2.1, we know that [latex]y_{1}=e^{t}[/latex], [latex]y_{2}=e^{-2t}[/latex], and [latex]y=c_{1}e^{t}+c_{2}e^{-2t}[/latex] are solutions of [latex]y''+y'-2y=0[/latex]. The choices of [latex]y_{1}[/latex], [latex]y_{2}[/latex] and [latex]y=c_{1}y_{1}+c_{2}y_{2}[/latex] is not by accident. It builds on section 2.2 Wronskian. 

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]. Then the family of the solutions [latex]y=c_{1}y_{1}+c_{2}y_{2}[/latex] with arbitrary coefficients [latex]c_{1}[/latex] and [latex]c_{2}[/latex] includes every solutions of the ODE if and only if there is a [latex]t=a[/latex] such that [latex]W(a)\neq0[/latex]. [latex]y=c_{1}y_{1}+c_{2}y_{2}[/latex] is called the general solution of the ODE and [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are called a fundamental set of solutions of the ODE.

Example 1: [latex]y_{1}=e^{3t}[/latex], [latex]y_{2}=te^{3t}[/latex]. Verify [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are solutions of [latex]y''-6y'+9y=0[/latex]. Do [latex]y_{1}[/latex] and [latex]y_{2}[/latex] form a set of fundamental solutions?

Exercise 1: [latex]y_{1}=e^{-t}[/latex], [latex]y_{2}=te^{-t}[/latex]. Verify [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are solutions of [latex]y''+2y'+y=0[/latex]. Do [latex]y_{1}[/latex] and [latex]y_{2}[/latex] form a set of fundamental solutions?

Example 2: [latex]y_{1}=e^{3t}\text{cos}(2t)[/latex], [latex]y_{2}=e^{3t}\text{sin}(2t)[/latex]. Verify [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are solutions of [latex]y''-6y'+13y=0[/latex]. Do [latex]y_{1}[/latex] and [latex]y_{2}[/latex] form a set of fundamental solutions?

Exercise 2: [latex]y_{1}=e^{-2t}\text{sin}(5t)[/latex], [latex]y_{2}=e^{-2t}\text{cos}(5t)[/latex]. Verify [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are solutions of [latex]y''+4y'+29y=0[/latex]. Do [latex]y_{1}[/latex] and [latex]y_{2}[/latex] form a set of fundamental solutions?

Example 3:  [latex]y=\text{sin}(3t)+i\text{cos}(3t)[/latex]. Verify both [latex]y_{1}=\text{cos}(3t)[/latex], [latex]y_{2}=\text{sin}(3t)[/latex] are also solutions of [latex]y''+9y=0[/latex].

Exercise 3: [latex]y=\text{cos}(2t)+i\text{sin}(2t)[/latex]. Verify both [latex]y_{1}=\text{sin}(2t)[/latex], [latex]y_{2}=\text{cos}(2t)[/latex] are also solutions of [latex]y''+4y=0[/latex]. We can use Wronskian to decide if a set of solution forms a fundament set of solutions. Unfortunately, this is like the chicken and egg problem, we may not be able to find [latex]y_{1}[/latex] and [latex]y_{2}[/latex] and check for the determinate. We need another way to check if there is a solution for the ODE or the IVP.

Theorem (Abel’s 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] where [latex]p(t)[/latex] and [latex]q(t)[/latex] are continuous on some open interval [latex]I[/latex]. Then the Wronskian is given by 

\[W(y_{1},y_{2})(t)=ce^{-\int p(t)dt}\]

where [latex]c[/latex] is depending on [latex]y_{1}[/latex] and [latex]y_{2}[/latex] but not on [latex]t[/latex].  Notice that [latex]W(y_{1},y_{2})(t)=ce^{-\int p(t)dt}[/latex] is either zero for all [latex]t[/latex] in the interval [latex]I[/latex] or is never zero in [latex]I[/latex].

Example 4: Find Wronskian of [latex]t^{3}y''-t^{2}(t+2)y'+(t-4)y=0[/latex]. 

Exercise 4: Find Wronskian of [latex]\text{sin}(t)y''+(\text{cos}(t))y'+(t+3)y=0[/latex]. 

Group Work

1. If [latex]t^{2}y''-y'+t^{2}\text{cos}(t)y=0[/latex] has [latex]y_{1}[/latex] and [latex]y_{2}[/latex] as a fundamental set of solutions and if [latex]W(y_{1},y_{2})(1)=5[/latex], find [latex]W(y_{1},y_{2})(-2)[/latex].

2. [latex]y_{1}=e^{-3t}\text{cos}(4t)[/latex], [latex]y_{2}=e^{-3t}\text{sin}(4t)[/latex]. Verify [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are solutions of [latex]y''+6y'+25y=0[/latex]. Do [latex]y_{1}[/latex] and [latex]y_{2}[/latex] form a set of fundamental solutions?

3. Show that if [latex]p(t)[/latex] is a differential positive function, then the ODE [latex](p(t)y')'+q(t)y=0[/latex] has Wronskian, [latex]W(t)=c/p(t)[/latex]. 

4. Find Wronskian of [latex]t^{2}y''-(t-2)y'+(t-4)y=0[/latex].