6C (Differential Equations)

For

k \neq = m

, the required Euler-equation solution is

x (t) = \frac{t ^{k} - t ^{m}}{k ^{2} - m ^{2}} .

For

k = m > 0

, the limiting resonant form is

x (t) = \frac{t ^{m} l o g t}{2 m} .

Both extend continuously to zero and satisfy the two boundary conditions. If

k = m = 0

, integration gives

x = \frac{1}{2} (lo g t)^{2} + C lo g t + D,

which cannot have a finite zero limit at

t = 0

, so no such solution exists.

b

y = t z

and

u = (x, z)^{T}

. The system becomes

t u^{'} = (43 - 6 - 5) u + t (1 1) + (1 2) .

The matrix has eigenpairs

1, (2, 1)^{T}

and

- 2, (1, 1)^{T}

(- t /3 - 2, - 1)^{T}

. Therefore

x = 2 C_{1} t + C_{2} t^{- 2} - \frac{t}{3} - 2,

y = C_{1} t^{2} + C_{2} t^{- 1} - t .

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