# Thread: sorry, another differential equation

1. ## sorry, another differential equation

find a general solution to the differential equation

(2+x) dy/dx = 3y

i got the answer y= 6+3x+e^(3c)
but i don't think that makes much sense..

2. Originally Posted by holly123
find a general solution to the differential equation

(2+x) dy/dx = 3y

i got the answer y= 6+3x+e^(3c)
but i don't think that makes much sense..
The differential equation separates
i.e.

$\displaystyle \frac{dy}{y} = 3 \frac{dx}{x+2}$

so

$\displaystyle \ln |y| = 3 \ln |x +2| + c$

3. okay thats what i got but i didn't know how to solve for y. i know you have to inject e to cancel out ln
so would it be y= 3(2+x) + e^(3c)
im not sure

4. Originally Posted by danny arrigo
The differential equation separates
i.e.

$\displaystyle \frac{dy}{y} = 3 \frac{dx}{x+2}$

so

$\displaystyle \ln |y| = 3 \ln |x +2| + c$

So $\displaystyle e^{\ln |y|} = e^{3 \ln |x +2| + c}$ so

$\displaystyle e^{\ln |y|} = e^{\ln |x +2|^3 + c} = e^{\ln |x +2|^3} \cdot e^{c}$

giving

$\displaystyle y = k (x+2)^3$ where $\displaystyle k = e^c$

5. Hello, holly123!

Find a general solution to the differential equation: .$\displaystyle (2+x)\frac{dy}{dx} \:= \:3y$

i got the answer: .$\displaystyle y\:=\: 6+3x+e^{3c}$
but i don't think that makes much sense. . . . . no, it doesn't

Separate the variables: .$\displaystyle \frac{dy}{y} \:=\:\frac{3\,dx}{x+2}$

Integrate: .$\displaystyle \int\frac{dy}{y} \:=\:3\int\frac{dx}{x+2} \quad\Rightarrow\quad \ln y \:=\:3\ln(x+2) + c$

. . $\displaystyle \ln y \:=\:\ln(x+2)^3 + \ln C \quad\Rightarrow\quad \ln y \:=\:\ln\bigg[C(x+2)^3\bigg]$

Therefore: .$\displaystyle y \:=\:C(x+2)^3$

6. thank you