sorry, another differential equation

• Jan 5th 2009, 12:21 PM
holly123
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..
• Jan 5th 2009, 12:25 PM
Jester
Quote:

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.

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

so

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

• Jan 5th 2009, 12:27 PM
holly123
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
• Jan 5th 2009, 12:31 PM
Jester
Quote:

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

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

so

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

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

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

giving

$y = k (x+2)^3$ where $k = e^c$
• Jan 5th 2009, 12:38 PM
Soroban
Hello, holly123!

Quote:

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

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

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

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

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

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

• Jan 5th 2009, 12:44 PM
holly123
thank you