Mathematics

Mathematics

Problem

The functions y=C1e2x+C2xe2xy = C_{1}e^{2x} + C_{2}x e^{2x} form a two-parameter family of curves. Which of the following second-order differential equations is satisfied by every member of this family and is of the least possible order? a) d2ydx2+4dydx+4y=0\displaystyle \frac{d^{2}y}{dx^{2}} + 4\frac{dy}{dx} + 4y = 0 b) d3ydx3−4d2ydx2+4dydx=0\displaystyle \frac{d^{3}y}{dx^{3}} - 4\frac{d^{2}y}{dx^{2}} + 4\frac{dy}{dx} = 0 c) d2ydx2−4dydx+4y=0\displaystyle \frac{d^{2}y}{dx^{2}} - 4\frac{dy}{dx} + 4y = 0 d) d2ydx2−2dydx+y=0\displaystyle \frac{d^{2}y}{dx^{2}} - 2\frac{dy}{dx} + y = 0

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