Answer to Question #167405 in Calculus for Rhon

Question

The points (-1,3) and (0,2) are on a curve, and at any point (x,y) on the curve (𝑑^2𝑦) /(𝑑𝑥^2) = 2 − 4𝑥. Find an equation of the curve.

Answer 1

Solution

If (𝑑^2𝑦) /(𝑑𝑥^2) = 2 − 4𝑥, then dy/dx = ∫(2-4x)dx = 2x-2x²+C

and y(x) = ∫(2x-2x²+C)dx = x²-2x³/3+Cx+D

Here C,D are arbitrary constants.

y(-1) = 1+2/3-C+D = 5/3-C+D = 3 => D-C = 4/3

y(0) = D = 2  => D = 2 and from previous equation C = 2-4/3 = 2/3

So the equation of the curve is y(x) = x²-2x³/3+2x/3+2

Answer

y(x) = x²-2x³/3+2x/3+2