A second–order linear homogeneous differential equation is given by y''-4y=0
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A second–order linear homogeneous differential equation is given by
y′′−4y=0.
Assume y is twice differentiable on the real line.
(a) Determine the general solution of the differential equation, clearly identifying all arbitrary constants.
(b) Substitute your result from part (a) back into the differential equation to justify that it indeed produces identically zero on the left–hand side.
(c) Given the initial conditions y(0)=2 and y′(0)=−1, determine the unique particular solution that satisfies both the differential equation and the initial conditions.
(d) Interpret, in a brief paragraph, the practical significance of the two arbitrary constants found in part (a) with respect to the dimension (order) of the differential equation.