Repeated roots and a forced equation

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

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Before you start

First order differential equations. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

$y''-2y'+y=6e^{x}$ , with $y(0)=0,\quad y'(0)=0$ .

Part a

Find the solution, including the choice of particular integral.

Part b

Find y(1) $e^{-1}$ .

Part c

Verify your solution by substituting it into the equation.

Model solution and review criteria

Part a

The auxiliary equation is (m−1)² = 0. Its complementary function is (A+Bx) $e^{1x}$ . Put y = u $e^{1x}$ ; the equation reduces to u″ = 6. Thus u = ( $\frac{6}{2}$ ) $x^{2}$ + Bx + A. The two initial conditions give A = B = 0.

Recognise the repeated root and both complementary terms.
Choose $x^{2}$ rather than x for the resonant forcing.
Apply both initial conditions.

Part b

The exponential cancels, leaving $\frac{6}{2}$ .

Evaluate the established solution.

Part c

For y = u $e^{1x}$ , y′ = (u′+1u) $e^{1x}$ and y″ = (u″+2u′+1u) $e^{1x}$ . Substitution cancels the u and u′ terms, leaving u″ $e^{1x}$ = 6 $e^{1x}$ .

Show both derivatives and cancellation.
Check the initial conditions as well as the equation.

Compare your own reasoning. No automatic examination marks are awarded.