Question

Assuming a, b ≠ 0, find the solution u = u(t) to the initial-value problem au′ = eᵃ u + bt, u(0) = b, in terms of a and b.

Answer 1

The solution involves separating variables of a first order linear differential equation, applying the integrating factor method, and integrating both sides to find the function u(t).

Step-by-step explanation:

To solve the initial-value problem au' = e^{a}u + bt, u(0) = b, we perform the separation of variables. First, divide by a to isolate u', which gives us:

u' = \frac{e^{a}u}{a} + \frac{bt}{a}

Now we have a first order linear differential equation. The integrating factor, \mu(t), is given by e^{\int P(t) dt}, where P(t) = \frac{e^{a}}{a}. Calculate the integrating factor:

\mu(t) = e^{\frac{e^{a}}{a}t}

Multiply the differential equation by the integrating factor:

e^{\frac{e^{a}}{a}t} u' - \frac{e^{2a}}{a} u = e^{\frac{e^{a}}{a}t} \frac{bt}{a}

Now we have a differential equation in the form:

(\mu(t)u)' = \mu(t) Q(t)

Where Q(t) = \frac{bt}{a}. Integrate both sides with respect to t to find u.