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 $y_1=e^t$, $y_2=e^{-2t}$, and $y=c_1{e}^t+c_2{e}^{-2t}$ are solutions of $y^{″}+y^{\prime }-2y=0$. The choices of $y_1$, $y_2$ and $y=c_1{y}_1+c_2{y}_2$ is not by accident. It builds on section 2.2 Wronskian. 

Theorem: Suppose $y_1$ and $y_2$ are solutions of the ODE, $y^{″}+p(t)y^{\prime }+q(t)y=0$. Then the family of the solutions $y=c_1{y}_1+c_2{y}_2$ with arbitrary coefficients $c_1$ and $c_2$ includes every solutions of the ODE if and only if there is a $t=a$ such that $W(a)\ne 0$. $y=c_1{y}_1+c_2{y}_2$ is called the general solution of the ODE and $y_1$ and $y_2$ are called a fundamental set of solutions of the ODE.

Example 1: $y_1=e^{3t}$, $y_2=t{e}^{3t}$. Verify $y_1$ and $y_2$ are solutions of $y^{″}-6y^{\prime }+9y=0$. Do $y_1$ and $y_2$ form a set of fundamental solutions?

Exercise 1: $y_1=e^{-t}$, $y_2=t{e}^{-t}$. Verify $y_1$ and $y_2$ are solutions of $y^{″}+2y^{\prime }+y=0$. Do $y_1$ and $y_2$ form a set of fundamental solutions?

Example 2: $y_1=e^{3t}\text{cos}(2t)$, $y_2=e^{3t}\text{sin}(2t)$. Verify $y_1$ and $y_2$ are solutions of $y^{″}-6y^{\prime }+13y=0$. Do $y_1$ and $y_2$ form a set of fundamental solutions?

Exercise 2: $y_1=e^{-2t}\text{sin}(5t)$, $y_2=e^{-2t}\text{cos}(5t)$. Verify $y_1$ and $y_2$ are solutions of $y^{″}+4y^{\prime }+29y=0$. Do $y_1$ and $y_2$ form a set of fundamental solutions?

Example 3:  $y=\text{sin}(3t)+i\text{cos}(3t)$. Verify both $y_1=\text{cos}(3t)$, $y_2=\text{sin}(3t)$ are also solutions of $y^{″}+9y=0$.

Exercise 3: $y=\text{cos}(2t)+i\text{sin}(2t)$. Verify both $y_1=\text{sin}(2t)$, $y_2=\text{cos}(2t)$ are also solutions of $y^{″}+4y=0$. 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 $y_1$ and $y_2$ 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 $y_1$ and $y_2$ are solutions of the ODE, $y^{″}+p(t)y^{\prime }+q(t)y=0$ where $p(t)$ and $q(t)$ are continuous on some open interval $I$. Then the Wronskian is given by 

$$W(y_1,y_2)(t)=c{e}^{-\int p(t)dt}$$

where $c$ is depending on $y_1$ and $y_2$ but not on $t$.  Notice that $W(y_1,y_2)(t)=c{e}^{-\int p(t)dt}$ is either zero for all $t$ in the interval $I$ or is never zero in $I$.

Example 4: Find Wronskian of $t^3{y}^{″}-t^2(t+2)y^{\prime }+(t-4)y=0$. 

Exercise 4: Find Wronskian of $\text{sin}(t)y^{″}+(\text{cos}(t))y^{\prime }+(t+3)y=0$. 

Group Work

1. If $t^2{y}^{″}-y^{\prime }+t^2\text{cos}(t)y=0$ has $y_1$ and $y_2$ as a fundamental set of solutions and if $W(y_1,y_2)(1)=5$, find $W(y_1,y_2)(-2)$.

2. $y_1=e^{-3t}\text{cos}(4t)$, $y_2=e^{-3t}\text{sin}(4t)$. Verify $y_1$ and $y_2$ are solutions of $y^{″}+6y^{\prime }+25y=0$. Do $y_1$ and $y_2$ form a set of fundamental solutions?

3. Show that if $p(t)$ is a differential positive function, then the ODE $(p(t)y^{\prime }{)}^{\prime }+q(t)y=0$ has Wronskian, $W(t)=c/p(t)$. 

4. Find Wronskian of $t^2{y}^{″}-(t-2)y^{\prime }+(t-4)y=0$.