Let satisfy the differential equation for all real

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

Problem

Let

y=y(x)

satisfy the differential equation

\dfrac{dy}{dx}=3\lvert x\rvert-4

for all real

x

. The graph of

y

is required to pass through the point

(1,5)

and to be continuous on

(-\infty,\infty)

. The presence of

\lvert x\rvert

causes the antiderivative to have different forms on the regions

x<0

and

x>0

. Answer each part thoroughly. (a) Compute an antiderivative of

3\lvert x\rvert-4

x>0

and a separate antiderivative on

x<0

. Write each result with its own constant of integration. (b) Use the point

(1,5)

to determine the constant for the branch defined on

x\ge 0

, and state the resulting expression

y_{+}(x)

valid for

x\ge 0

. (c) Impose continuity at

x=0

to solve for the constant in the branch defined on

x\le 0

, obtaining

y_{-}(x)

valid for

x\le 0

. (d) Combine your results to write a single piecewise-defined formula for

y(x)

that satisfies the differential equation for all real

x

, the initial condition, and the continuity requirement. Justify that the final function is differentiable at

x=0

. (A formal proof is not required, but clear reasoning must be provided.)