Mathematics

Mathematics

Problem

Let dydx=ex+2\frac{dy}{dx} = e^{x} + 2 for all real xx. Point PP is (0,1)(0,1) and point QQ is (1,4)(1,4). a. Part 1: Find the unique particular solution yP(x)y_P(x) that passes through PP. b. Part 2: Determine whether QQ lies on the curve yP(x)y_P(x). Provide complete justification. c. Part 3: Prove that no solution curve passing through QQ can intersect the curve yP(x)y_P(x) for any real value of xx.
Intuitive explanation

You're looking at a differential equation where you know the slope function everywhere—the key is to reverse-engineer the original function by integrating. Since dydx\frac{dy}{dx} tells you the rate of change, integrating it gives you yy itself, plus an unknown constant. Use the initial condition at point PP to pin down that constant, then check whether point QQ actually lies on your curve.

Intuitive explanation

You're looking at a differential equation where you know the slope function everywhere—the key is to reverse-engineer the original function by integrating. Since dydx\frac{dy}{dx} tells you the rate of change, integrating it gives you yy itself, plus an unknown constant. Use the initial condition at point PP to pin down that constant, then check whether point QQ actually lies on your curve.

Browse the problem library