2.4 Differences between Linear and Non-linear

• Existence:

  • does a solution exist?

• uniqueness

  • is the solution unique? (only one solution?)

• solvability

  • whether all solutions are solvable

Theorem EU-2: existence and uniqueness for non-linear ODEs

• let $f(t,y)$ and $\frac{\partial f}{\partial y}$

• what is the graphical solution for uniqueness

  • when there is an initial solution there only can be one curve through the point (they cannot intersect)

Examples

theorem EU-2

EXAMPLE 1$y\frac{dy}{dx}=x^2,y(0)=y_0$

$$\begin{array}{r}\text{ using the theorem we see that}\frac{dy}{dx}=f(x,y)\\ \frac{dy}{dx}=\frac{x^2}{y}\to f(x,y)\\ \int ydy=\int x^{2}dx\\ \frac{y^2}{2}=\frac{x^3}{3}+\frac{y_0^2}{2}\end{array}$$

Example 2 $y{y}^{\prime }=x^2$ case 1: $y_0=!0$

• there is a unique solution in an open interval about x=0
  • we must find the binding limit (where the function becomes undefined (vertical tangent)) so we just solve for that interval :)

• we have two solutions in (0,inf) satisfying y0=0

  • so there is no solutions in an open interval containing 0
  • there are 2 solutions on the interval (0, inf)