Let be a solution to the differential equation dy dx =1+y^3 with initial condition

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

Problem

Let yy be a solution to the differential equation dydx=1+y3\frac{dy}{dx}=1+y^3 with initial condition y(0)=0y(0)=0. For this differential equation it can be shown that d2ydx2=3y2(1+y3).\frac{d^2y}{dx^2}=3y^2\left(1+y^3\right). A table of Euler approximations will be constructed on the interval −0.5≤x≤0.5-0.5\le x\le0.5 with uniform step size h=0.25h=0.25, starting at x=0x=0. (a) Use the given formula for d2ydx2\dfrac{d^2y}{dx^2} to determine the sign of the second derivative for all xx in the interval −0.5≤x≤0.5-0.5\le x\le0.5. Explain your reasoning. (b) Without performing any arithmetic for the Euler table, state whether each Euler estimate on this interval will be an overestimate or an underestimate of the corresponding exact yy–value. Support your conclusion with a clear concavity argument.

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