Basic Separation of Variables problem

[obstinatus]

I'm self-teaching through Tenenbaum & Pollard's "Ordinary Differential Equations", and for some reason I'm completely stuck on one of the problems, Ch.2, lesson 6, problem #6:

Find a 1-parameter family of solutions for [...] the differential equation:

6) yx²dy-y³dx = 2x²dy.

I didn't have trouble with any of the previous problems, but the algebra is evading me here. The proffered solution is:

(cx + 1)y² = (y-1)x, x =/ 0, y =/ 0; y = 0.

but I can't find any families that don't involve fractions, let alone this one. The subsequent problems also seem to have an algebra trick that I'm missing, so once I understand this one I'll be fine I think.

 

[micromass]

If you see $x$ as the function with variable $y$, then it's a Bernouilli differential equation.

 

[chiro]

Hey obstinatus.

A lot of differential equations (particularly in undergraduate college/university courses) involve classification using known methods.

When you start doing this subject it's usually the case where you have to learn the families of differential equations and then apply the known techniques to get a solution.

A lot of introductory mathematics courses are like this and it will help you (I think) in your education if you realize this and utilize it.

 

[rahul_26]

Given equation,
$$y\ x^2 \cdot dy-y^3 \cdot dx=2\ x^2\ dy$$
We can rewrite the given equation as
$$\Big(y\ x^2-2\ x^2\Big)dy=y^3\ dx$$
$$\Rightarrow \frac{y-2}{y^3} \cdot dy=\frac{dx}{x^2}$$
Now you can integrate both sides and find the solution.

 

[obstinatus]

Hey obstinatus.

A lot of differential equations (particularly in undergraduate college/university courses) involve classification using known methods.

When you start doing this subject it's usually the case where you have to learn the families of differential equations and then apply the known techniques to get a solution.

A lot of introductory mathematics courses are like this and it will help you (I think) in your education if you realize this and utilize it.

This was not helpful.

 

[chiro]

It's just how it is in a lot of mathematics education.

You don't have the general formulas you might otherwise think exists in mathematics and the reality is that a lot of classification exists to solve particular kinds of problems and not just all the general ones you probably think are easily solved.

It's important to know the realities of how things work and be aware of the effects they have.