Help. How to recognize the different types of first order Differential equations EASILY? The separable, homo, exact, non exact, integrating factor, first order linear de? Also the second order de? Thank you

Question

sepeario · Answer

attachment

sepeario · Answer

http://www.math.hawaii.edu/~lee/calculus/DE.pdf

anonymous · Answer

To add to the info listed in the pdf above, you can check to see if a first-order equation is homogeneous if it satisfies the following.

Given the ODE $\dfrac{dy}{dx}=f(x,y)$, the ODE is homogeneous if for some real $n$,
$$f(tx,ty)=t^nf(x,y)$$
For example, consider the homogeneous ODE given in the document:
$$y'=\frac{y^3-4xy^2}{x^3+8x^2y}=f(x,y)$$
You have
$$\begin{align*}
f(tx,ty)&=\frac{t^3y^3-4t^3xy^2}{t^3x^3+8t^3x^2y}\\
&=\frac{y^3-4xy^2}{x^3+8x^2y}\\
&=t^0f(x,y)
\end{align*}$$

anonymous · Answer

Thank you so much guys