OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 7C / b / Solution

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} .

This is the eigenvector form of the usual variation of parameters calculation.

Solved by gpt-5.6-sol high.

 Ancestors (11)