# Thread: Solving for Original Equation

1. ## Solving for Original Equation

Question: Solve for the following differential equation for the original equation:

$\displaystyle y' = \frac{7-4x}{4}$

Is my working correct?

$\displaystyle \frac{dy}{dx} = \frac{7-4x}{4}$

$\displaystyle 4dy = (7-4x)dx$

$\displaystyle \int 4dy = \int 7dx - \int 4xdx$

$\displaystyle 4y = 7x - \frac{4x^2}{2} + C$

$\displaystyle 8y = 14x - 4x^2 + 2C$ (multiply both sides by 2 to get rid of the half)

$\displaystyle 8y - 14x + 4x^2 - 2C = 0$

2. Yes, although how you didn't see 4/2 = 2 is a mystery. I'd have cancelled in the third line rather than multiplying through by 2.

2C and -2C are both constants and the convention is to either define a new constant or absorb the 2

3. Originally Posted by sparky
Question: Solve for the following differential equation for the original equation:

$\displaystyle y' = \frac{7-4x}{4}$

Is my working correct?

$\displaystyle \frac{dy}{dx} = \frac{7-4x}{4}$

$\displaystyle \int 4dy = \int 7dx - \int 4xdx$

$\displaystyle 4y = 7x - \frac{4x^2}{2} + C$

$\displaystyle 8y = 14x - 4x^2 + 2C$

$\displaystyle 8y - 14x + 4x^2 - 2C = 0$
Yes your working is correct. But your answer could be further simplified by dividing the whole equation by 2.

$\displaystyle 4y=7x-\frac{4x^2}{2}+C$

$\displaystyle 4y=7x-2x^2+C$

$\displaystyle y=\frac{1}{4}\left(7x-2x^2+C\right)$

$\displaystyle 4y = 7x - \frac{4x^2}{2} + C$
$\displaystyle 4y - 7x + 2x^2 - C$