“Grouping” differential equations and the appropriate methods for solving them

I have an exam coming up and a pretty large part is going to be on first-order ordinary differential equations. This has been my weakest subject during the semester so I am trying to get down the basics. In particular, I am trying to classify the different types of ordinary DE's and the methods for solving them. Below, I am going to write down the different types of equations and how I would solve them. I would really appreciate if you guys could point out if I am using a wrong approach or if I have completely misunderstood something. (I hope this is a valid question for stackexchange, if not, I kindly ask one of the moderators to remove it).

ps: I apologize for the length of this question. It turned out longer than I expected.

First-order homogenous linear differential equation

Diff. equations in the form of:

$$y'(x)+g(x)y(x)=0$$

Method for solving: Separation of variables

First-order inhomogeneous linear differential equation

Diff. equations in the form of:

$$y'(x)+g(x)y(x)=r(x), \space \space \space r(x)\not=0$$

Method for solving:

1. Solve the homogenous system: $y'(x)+g(x)y(x)=0$ and find the homogenous solution $y_h(x)$
2. Use "variation of parameters" to find a particular solution $y_p(x)$
3. The general solution is $y(x)=y_h(x)+y_p(x)$

First-order nonlinear differential equation

Diff. equations in the form of:

$$y'(x)=f(x,y)$$

The "method for solving" depends on whether or not the equation is separable.

In case the equation is separable then I can just use "separation of variables" and write the equation in the form: $y'(x)=f(x,y)=g_1(x)\cdot g_2(y)\iff \frac{dy}{dx}=g_1(x)\cdot g_2(y) \iff \frac{dy}{g_2(y)}=g_1(x)dx$.

The only nonlinear non-separable diff. equations I need to know are Bernoulli differential equations and equations in the form of $y'=F(\frac{y}{x})$

Bernoulli differential equation

Diff. equation in the form of:

$$y'(x)+p(x)y(x)=q(x)(y(x))^n$$

As far as I understand, for $n=0,1$ this becomes a linear differential equation and I can solve it like I solved the previous diff. equations.

Method for solving for $n \ge 2$:

1. Divide both sides by $y^n \implies \frac{y'}{y^n}+p(x)y^{1-n}=q(x)$
2. Substitute $v=y^{1-n}$ to convert into a diff. equation in terms of "$v$".
3. To determine what $y'$ is, differentiate $v$. $\implies v'=(1-n)y^{-n}y' \iff \frac{y'}{y^n}=\frac{v'}{(1-n)}$
4. The differential equation becomes: $\frac{v'}{1-n}+p(x)v=q(x)$
5. Solve by using "separation of variables" and "variation of parameters".
6. Resubstitute

Differential equation: $y'=F(\frac{y}{x})$

Method for solving:

1. Substitute $v(x)=\frac{y}{x} \iff y=xv; \space y'=v+xv'$
2. The equation becomes $v+xv'=F(v) \iff xv'=F(v)-v$
3. This equation is now separable and can be solved by the methods I mentioned above.
• Looks good for me. Differential equations of form $y' = F(\frac{y}{x})$ are also called homogeneous. – Evgeny Jul 15 '15 at 16:16
• @Evgeny Thanks! – qmd Jul 15 '15 at 16:31
• you're missing exact equations. Maybe this wasn't covered yet? – James S. Cook Jul 15 '15 at 17:18
• @JamesS.Cook You're right, we haven't covered these yet. Same with $n$-th order linear diff. equations. – qmd Jul 15 '15 at 18:41
• Very well then, you'll see them soon enough. Best wishes on your exam. – James S. Cook Jul 15 '15 at 18:46