Final answer:
The solution to the initial value problem with a delta function is a piecewise function involving exponentials and a step function. It shows behavior change at the impulse point, and the solution is found using integrating factors and initial conditions.
Step-by-step explanation:
The student asks for a solution to the initial value problem y'+y = 7+δ(t-1), with y(0) = 0, where δ(t-1) represents a delta function at t = 1. This is a first-order linear differential equation with an impulse forcing function due to the delta function.
To solve this initial value problem, we can use the integrating factor method. The integrating factor μ(t) is given by μ(t) = e∫ p(t) dt, where p(t) = 1 in this case.
Therefore, the integrating factor is simply et. Multiplying the entire differential equation by this integrating factor yields ety' + ety = 7et + etδ(t-1). Now, the left side of the equation is the derivative of ety with respect to t.
Integral of both sides then gives ety = 7et + H(t-1) + C, where H(t-1) is the Heaviside step function and C is the constant of integration. The initial condition y(0) = 0 leads to C = -7. Finally, by dividing both sides by et, we obtain the solution y(t) = 7 + H(t-1)e1-t - 7e-t.
The presence of the Heaviside step function indicates the change in the solution behavior at t = 1 due to the impulse from the delta function.