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 = (-1)/(2)\\B = (-1)/(2)\\C = (-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 = (-1)/(2)`

From 2.

`2A = 2B\\B = A\\B = (-1)/(2)`

From 3.

`2A + B - 2C = 0\\2A + A - 2C = 0\\3A - 2C = 0\\C = (3A)/(2), A = -(1)/(2)\\C = -(3)/(4)`