3.2 The Wronskian and Superposition

Existence and Uniqueness Theorem

• $y^{\prime \prime }+p(t)y^{\prime }+q(t)y=g(t),y(t_0)y_0,y^{\prime }(t_0)=y_0^{\prime }$

  • where p,q and g are continuous on an open interval and it contains the point $t_0$,

    • that means the problem has exactly one solution $y=\varphi (t)$ and the solution exists in the same interval

• What does this mean?

  • 1. the IVP must have a solution (a solution exists)
  • 2. the IVP has only one solution (the solution is unique)
  • 3. the solution $\varphi$ is defined throughout the interval I where everything is continuous and 2x differentiable

• Example

  • find the longest interval in which the solution of the IVP is certain to exist

    • $(t^2-3t)y^{\prime \prime }+t{y}^{\prime }-(t+3)y=0,$ $y(1)=2$ $y^{\prime }(1)=1$
  • if the equation is written of the form as the definition
    • $p(t)=\frac{1}{t-3},q(t)=-\frac{t+3}{t(t-3)}$ and $g(t)=0$
    • we can see there are discontinuities at t=0 and 3, therefore the longest open interval the solution can exist in is $0<t<3$

The Principle of Superposition

• when you have 2 solutions, a linear combination of the 2 will also be a solution