Consider the differential equation

MATHEMATICS · Single correct · CBSE · Mathematics · Differential Equations · Differential Equations

Problem

Consider the differential equation dydx=ex\dfrac{dy}{dx}=e^{x}. Determine whether a single particular solution can pass through both points (0,2)(0,2) and (2,10)(2,10). (a) Yes, because e2≈8e^{2}\approx 8, so y(2)=2+8=10y(2)=2+8=10. (The numbers work out.) (b) Yes, because introducing a constant of integration can always make the solution pass through any two points. (c) No, because the particular solution through (0,2)(0,2) is y=ex+1y=e^{x}+1, giving y(2)≈8.39≠10y(2)\approx 8.39\neq 10. (The point (2,10)(2,10) is off the curve.) (d) No, because taking y=xex+Cy=xe^{x}+C as the antiderivative leads to y(2)≈16.8y(2)\approx 16.8, which is greater than 1010. (So the solution cannot pass through both points.)

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