Consider the first–order differential equation dy dx =3y
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Consider the first–order differential equation
dxdy=3y.
A one–parameter family of functions is proposed as
y(x)=Ce3x, where C is a real constant.
(a) Verify that every member of the family y(x)=Ce3x satisfies the differential equation.
(b) Give a reasoned explanation, in complete sentences, of why the verification in part (a) implies that the differential equation possesses infinitely many distinct solutions.
(c) Describe, without computation, how the order of a differential equation limits the maximum possible number of arbitrary constants in its general solution. Relate your discussion specifically to the first–order equation given above.