(a) specify a suitable solution that will reduce a differential equation of the form dy/dx = f(x/y) to a separable equation. derive an expression for dy/dx using your substitution.

Question

z4k4r1y4 · Answer

(b) apply this technique to find the general solution for:
$$\frac{ dy }{ dx}=\frac{ 3x^2+8y^2 }{ 4xy}$$
giving your answer in the form y^2=g(x)

christos · Answer

You could solve the expression with respect to y. And because y is to the power of 2 , you will get two solutions , a positive and a negative one. Then you could integrate those. Or you could approach this as an inverse implicit differentiation.

z4k4r1y4 · Answer

@ganeshie8  any suggestions for part (a)?

ganeshie8 · Answer

Let $y=vx$

ganeshie8 · Answer

plugging that in the right hand side, $f(x/y)$, eliminates $y$ completely : 

 $$f(\frac{x}{y}) = f(\frac{x}{vx}) = f(1/v)$$

ganeshie8 · Answer

$y=vx$

differentiating both sides with respect to $x$, we get :
$\dfrac{dy}{dx} = \dfrac{dv}{dx}*x+v*1$

ganeshie8 · Answer

In the given differential equation, you can replace left hand side derivative with above expression

ganeshie8 · Answer

$\dfrac{dy}{dx} = f(\frac{x}{y})$

changes to

$ \dfrac{dv}{dx}*x+v*1 = f(1/v)$

which is separabel

z4k4r1y4 · Answer

yes, i understand what you've done. thank you, that was very helpful.

ganeshie8 · Answer

you may find useful this video on substitution methods http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-4-first-order-substitution-methods/