Question

The equation y ′′ + y ′ − 2y = x 2 is called a differential equation because it involves an unknown function y and its derivatives y ′ and y ′′. Find constants A, B, and C such that the function y = Ax2 + Bx + C satisfies this equation. (Differential equations will be studied in detail in Calculus 2).

Answer 1

Answer: $A = \frac{-1}{2}\\B = \frac{-1}{2}\\C = \frac{-3}{4}$

Step-by-step explanation:

First we derive y two times $y = Ax^{2} + Bx + C\\y' = 2Ax + B\\y'' = 2A$

Second we replace in the differential equation y' and y'':

$2A + 2Ax + B - 2(Ax^{2} + Bx + C) = x^{2}$

grouping terms

$(-2A)x^{2} + (2A-2B)x + (2A + B -2C) = x^{2}$

The terms on the left side must be equal to the terms of the right side, so

$1. (-2A) x^{2} = x^{2} \\2. (2A - 2B)x = 0\\3. (2A + B - 2C) = 0$

From 1.

$A = \frac{-1}{2}$

From 2.

$2A = 2B\\B = A\\B = \frac{-1}{2}$

From 3.

$2A + B - 2C = 0\\2A + A - 2C = 0\\3A - 2C = 0\\C = \frac{3A}{2}, A = -\frac{1}{2}\\C = -\frac{3}{4}$