Summary

Chapter recap.

1. A differential equation is any equation containing at least one derivative of an unknown function — ordinary or partial.
2. The order of a differential equation is the highest derivative present.
3. If a differential equation is expressible in polynomial form, the integral power to which the highest-order derivative is raised is called the degree.
4. If the equation cannot be expressed in polynomial form with the highest-order derivative as the leading term, the degree is not defined.
5. An equation containing only ordinary derivatives of a function of a single independent variable is an ordinary differential equation (ODE).
6. An equation involving only partial derivatives of a function of two or more independent variables is a partial differential equation (PDE).
7. Eliminating one arbitrary constant yields a first-order differential equation; eliminating two yields a second-order equation; and so on.
8. A solution of a differential equation is an expression for the dependent variable in terms of the independent variable(s) that satisfies it.
9. The solution containing as many arbitrary constants as the order is the general solution.
10. Assigning particular values to those constants gives a particular solution.
11. An equation of the form $f_1(x)g_1(y)\,dx+f_2(x)g_2(y)\,dy=0$ is a variables-separable (or simply separable) equation.
12. A function $f(x,y)$ is homogeneous of degree $n$ if $f(tx,ty)=t^nf(x,y)$ for suitably restricted $x,y,t$ — Euler's homogeneity.
13. If $f(x,y)$ is homogeneous of degree zero, it can always be written as $g\!\left(\dfrac{y}{x}\right)$.
14. An ODE is in homogeneous form if written as $\dfrac{dy}{dx}=g\!\left(\dfrac{y}{x}\right)$.
15. $M(x,y)\,dx+N(x,y)\,dy=0$ is homogeneous exactly when $M$ and $N$ are homogeneous functions of the same degree.
16. A first-order linear equation $\dfrac{dy}{dx}+Py=Q$ ($P,Q$ functions of $x$ only, no product of $y$ with $y'$, both to the first degree) is solved by $y\,e^{\int P\,dx}=\displaystyle\int Q\,e^{\int P\,dx}\,dx+C$, where $e^{\int P\,dx}$ is the integrating factor (I.F.). …