Solving y'' + 4y = e^(-t), y(0) = 1, y'(0) = 0

Differential Equations · Axiom Academy

EXAMPLE Solving y'' + 4y = e^ -t with Laplace Transforms Work a forced initial value problem end to end: transform, solve algebraically, and invert. Solve y'' + 4y = e^ -t subject to y(0) = 1 and y'(0) = 0 using the Laplace transform method. Nice work — you solved a forced second-order IVP with Laplace transforms from start to finish. Transform first: Apply the Laplace transform to both sides to turn the differential equation into an algebraic one in Y(s) . Initial conditions matter: carries the ICs right into the transformed equation — substitute them immediately. Combine before decomposing: Write the right side as ONE fraction before doing partial fractions — a slip here (like the classic s(s+1)+1 arithmetic) silently corrupts every step downstream. Match the factor type: A linear factor (s+1) gets ; an irreducible quadratic (s^2+4) needs the full — never just . Result: — check it by confirming y(0)=1 and y'(0)=0 . This same four-move pattern — transform, solve algebraically, decompose, invert — handles any linear IVP with constant coefficients, no matter the forcing function.

This is the written version of the interactive lesson above. See the full Differential Equations course.