1.1 Basic ODEs and Direction Fields

Slope Fields

• Isocline

  • A curve in the t-y plane on which the slope is constant

  • Ex

    • $\frac{dy}{dx}=x-y^2$ we can find the isocline to find these curves and we can use those to find the plo

• Unstable and Stable

  • unstable is when the slope is increasing away from equilibrium

  • Stable is when the slope is decreasing to the equilibrium

Analytical solution

• Test a solution-

  • Its easy to verify a solution

    • We just need to differentiate and verify

  • EX

    • check that $P(t)=C{e}^{rt}$ is a solution for $P^{\prime }=rP$
      • ⇒ $r(C{e}^{rt})=rP(t)⟹rP(t)=rC{e}^{rt}$

• Derive a solution

  • this is the DE :)

  • Ex

    • Derive the solution of $P^{\prime }=rP$
      • 2.2 Separable DES we just separate it and solve it