7C (Differential Equations)

det (A - λ I) = λ (λ - 1) .

For

λ_{1} = 0

and

λ_{2} = 1

, corresponding eigenvectors are

u_{1} = (21), u_{2} = (31) .

They are linearly independent, so the general homogeneous solution is

z (t) = α (21) + β e^{t} (31) .

b

Resolve the forcing in the eigenbasis:

(1 a) = (3 a - 1) u_{1} + (1 - 2 a) u_{2} .

In the zero-eigenvalue direction, a constant forcing produces the particular integral

(3 a - 1) t u_{1}

. In the unit-eigenvalue direction, a constant particular integral is

- (1 - 2 a) u_{2}

. Thus a particular integral depends on time exactly when

a \neq = \frac{1}{3} .

Adding the complementary solution from part (a), the general solution is

z (t) = α u_{1} + β e^{t} u_{2} + (3 a - 1) t u_{1} - (1 - 2 a) u_{2} .

For either eigenpair above,

λ_{i} \in {0, 1}

and hence

λ_{i}^{n} = λ_{i}

n

. Therefore

\frac{d ^{n}}{d t ^{n}} (u_{i} e^{λ_{i} t}) = λ_{i}^{n} u_{i} e^{λ_{i} t} = A u_{i} e^{λ_{i} t},

so both terms in the solution from part (a), and every linear combination of them, solve the higher-order system.

n

2 n

-dimensional solution space. The displayed family supplies two independent solutions, so there must be

2 n - 2

further linearly independent solutions.

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