PX153 - D2 - solving 2nd order homogeneous ODEs

derivation

consider

\frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} + 2 y = 0

we postulate the solution:

y = A e^{λ x}

λ^{2} A e^{λ x} + λ A e^{λ x} + 2 A e^{λ x} = 0 $ $ $ $ λ^{2} y + λ y + 2 y = 0

looking for a solution across a range of y:

⟹ λ^{2} + λ + 2 = 0

- this is the auxiliary equation, an algebraic equation that can be solved to find $λ$ , which has 2 solutions:
$$\lambda_{1}=- \frac{1}{2}(1+\sqrt 7 i)$$$$\lambda_{2}= - \frac{1}{2}(1-\sqrt 7 i)$$

the general solution contains all possible solutions:
$$y(x)= e^{\lambda_{1}x}+e^{\lambda_{2}x}$$
for a particular solution, we have to apply boundary conditions
- eg: if $y (x = 0) = 0$ and $\frac{d y}{d x} |_{x = 0} = 1$
  $y (x) = C e^{λ_{1} x} + D e^{λ_{2} x}$
  $y (0) = C + D = 0 ⟹ C = - D$
  $\frac{d y}{d x} = λ_{1} C e^{λ_{1} x} + λ_{2} D e^{λ_{2} x}$
  at $x = 0 : \frac{d y}{d x} = λ_{1} C + λ_{2} D = 1$
  $⟹ (λ_{1} - λ_{2}) C = 1$
  $λ_{1} = - \frac{1}{2} (1 + \sqrt{7} i); λ_{2} = - \frac{1}{2} (1 - \sqrt{7} i)$
  $⟹ \sqrt{7} i C = 1$
  $⟹ C = - D = \frac{- 1}{\sqrt{7} i}$
  $∴ y (x) = \frac{- 1}{\sqrt{7} i} e^{- \frac{x}{2}} (e^{i \frac{\sqrt{7} i}{2}} + e^{- i \frac{\sqrt{7} i}{2}})$
  $= \frac{- 2}{\sqrt{7}} e^{- \frac{x}{2}} (\sin \frac{\sqrt{7}}{2} x)$
for an nth order ODE, the auxiliary equation will be an nth order polynomial. we need to find n linearly independent solutions, but we can have repeated roots
- eg: for $\frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} + y = 0$
  - the auxiliary equation: $l^{2} + 2 l + 1 - 0$
    $(l + 1)^{2} = 0$
    $l = - 1$
  - we only found one solution: $y (x) = A e^{l x} = A e^{- x}$
  - in the case of repeated roots, if we have one solution, $y_{1} = A e^{l x}$ , then try $y_{2} = B x e^{l x}$ (multiply by the independent variable)
  - working through this:
    $\frac{d y_{2}}{d x} = B e^{l x} + l B x e^{l x}$
    $\frac{d^{2} y_{2}}{d x^{2}} = B l e^{l x} + B l e^{l x} + B l^{2} x e^{l x} = 2 B l e^{l x} + B l^{2} x e^{l x}$
  - show that this satisfies $\frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} + y = 0$
    $- 2 B e^{- x} + B x e^{- x} + 2 B e^{- x} - 2 B x e^{- x} + B x e^{- x} = 0$

summary

for a 2nd order linear homogeneous ODE:
- try solution of the form $A e^{λ x}$
- find the auxiliary equation (quadratic in $λ$ )
- if there are two distinct roots, GS: $y = A e^{λ_{1} x} + B e^{λ_{2} x}$
- if there are repeated roots, GS: $y = A e^{λ x} + B x e^{λ x}$
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