MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A function y=y(x) satisfies dxdy=x5 and the initial condition y(−2)=7.
Because the integrand involves x1, absolute values arise in the antiderivative.
Respond completely to each part.
(a) Find the general solution of the differential equation, expressing your answer so that the absolute value required by the antiderivative is evident.
(b) Determine the particular solution passing through (−2,7). Rewrite your result without an absolute-value symbol when possible, and explain why the rewrite is legitimate.
(c) Specify the maximal domain of validity of your particular solution, making clear why no larger domain is possible.
(d) Verify by direct differentiation that your final expression satisfies the original differential equation on the domain stated in part (c).