An unsignposted first-order initial-value problem

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

$y'-y=3e^{x}$ , with $y(0)=-2$ .

Part a

Solve the differential equation, showing your method.

Part b

Find y′(0).

Part c

Explain why a particular integral of the form K $e^{rx}$ , with r the stated coefficient, fails.

Model solution and review criteria

Part a

An integrating factor is $e^{-1x}$ . It gives (y $e^{-1x}$ )′ = 3, hence y = (3x + A) $e^{1x}$ . The initial condition gives A = -2. A resonant trial 3x $e^{1x}$ is an alternative with justification.

Choose and justify an integrating factor or resonant trial.
Integrate and include a constant.
Apply the initial condition.

Part b

The original equation gives y′(0) = 1(-2) + 3 = 1. This checks the completed solution.

Use the initial state in the equation.

Part c

That exponential solves the homogeneous equation, so substitution gives zero for every constant K. A factor of x is needed to produce the nonzero forcing.

Demonstrate cancellation rather than naming resonance alone.

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