How Can I Solve This Challenging Bernoulli Equation?

[cragar]

Homework Statement

Ay'+Bxy=Cy
y=f(x)
A,B,C are real constants

The Attempt at a Solution

This kinda looks like a Bernoulli equation but not really.
I thought about using an integrating factor but there is function of x on the right side.
If I tried undetermined coefficients what would my guess function be.

 

[elvishatcher]

It seems like you could solve this by first manipulating to get $$y' + \frac{Bx-C}{A}y = 0$$ at which point you now have an ODE of the form $$y' + P(x)y = Q(x)$$ and there is a general way to solve such ODEs.

 

[cragar]

where Q(x)=0 and then use an integrating factor.

 

[elvishatcher]

Yep, seems like that oughta work

 

[cragar]

if you do that you get y=0.

 

[ehild]

It seems like you could solve this by first manipulating to get $$y' + \frac{Bx-C}{A}y = 0$$

or $$y' = -\frac{Bx-C}{A}y$$

which is separable. $$\frac{y'}{y} = -\frac{Bx-C}{A}$$.

ehild

 

[cragar]

wow can't believe I missed that , thanks for the help
ok so I would get
$ln(y)= \frac{-1}{A}(\frac{Bx^2}{2}-Cx)+F$
F= integration constant
then I just raise each side to e and I will have y

 

[ehild]

Exactly. It will be a bit simpler if you eliminate the minus sign in front of the parentheses,

$$\ln(y)=\frac{1}{A}(Cx-B\frac{x^2}{2})+F$$

ehild