Two differential equations are presented for comparison

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

Two differential equations are presented for comparison. I. dydx=y\dfrac{dy}{dx}=y II. d2ydx2−y=0\dfrac{d^{2}y}{dx^{2}}-y=0 Both are defined for sufficiently differentiable functions on R\mathbb R. (a) Solve Equation I and state explicitly the number of arbitrary constants in its general solution. (b) Solve Equation II and state explicitly the number of arbitrary constants in its general solution. (c) Provide a rigorous argument, rooted in the structure of successive differentiation, that an nnth–order linear homogeneous differential equation with constant coefficients generally admits nn arbitrary constants in its solution family (assuming distinct characteristic roots). (d) Explain how the availability of nn arbitrary constants guarantees that a set of nn independent initial conditions (for example, y(0),y′(0),…,y(n−1)(0)y(0),y'(0),\dots ,y^{(n-1)}(0)) is sufficient to determine a unique member of the family obtained in part (c).

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