Order and degree
Order \(=\) order of the highest derivative present. Degree \(=\) power of that highest derivative once the equation is polynomial in all derivatives (free of fractional powers, and with no derivative inside a transcendental function). If it cannot be made polynomial in the derivatives, the degree does not exist.
Forming an equation
Number of independent arbitrary constants \(=\) order of the resulting differential equation. Differentiate the family that many times and eliminate the constants.
Variables separable
\[ g(y)\,dy = f(x)\,dx \quad\Longrightarrow\quad \int g(y)\,dy = \int f(x)\,dx + C. \]
Reducible to separable
\[ \dfrac{dy}{dx} = f(ax + by + c), \quad z = ax + by + c \;\Longrightarrow\; \dfrac{dz}{dx} = a + b\,f(z). \]
Homogeneous equation
\[ \dfrac{dy}{dx} = F\!\left(\dfrac{y}{x}\right), \quad y = vx \;\Longrightarrow\; v + x\dfrac{dv}{dx} = F(v), \]
which separates into \( \dfrac{dv}{F(v) - v} = \dfrac{dx}{x} \).
Linear equation in \(y\)
\[ \dfrac{dy}{dx} + P(x)\,y = Q(x), \qquad \text{I.F.} = e^{\int P\,dx}, \]
\[ y\cdot e^{\int P\,dx} = \int Q\,e^{\int P\,dx}\,dx + C. \]
Linear equation in \(x\)
\[ \dfrac{dx}{dy} + P(y)\,x = Q(y), \qquad \text{I.F.} = e^{\int P\,dy}, \]
\[ x\cdot e^{\int P\,dy} = \int Q\,e^{\int P\,dy}\,dy + C. \]
Standard integrals that recur
\[ \int \dfrac{dx}{x} = \ln|x|, \qquad \int \dfrac{dx}{\sqrt{1-x^{2}}} = \sin^{-1}x, \qquad \int e^{x}\,dx = e^{x}, \]
\[ \int a^{x}\,dx = \dfrac{a^{x}}{\ln a}, \qquad \int \dfrac{dx}{1+x^{2}} = \tan^{-1}x. \]
Application models
\[ \text{Growth:}\quad \dfrac{dx}{dt} = kx \;\Longrightarrow\; x = x_{0}\,e^{kt} \quad (k>0). \]
\[ \text{Decay:}\quad \dfrac{dA}{dt} = -kA \;\Longrightarrow\; A = A_{0}\,e^{-kt} \quad (k>0). \]
\[ \text{Newton cooling:}\quad \dfrac{dT}{dt} = k\,(T - T_{m}) \quad (k<0). \]
\[ \text{Mixture:}\quad \dfrac{dx}{dt} = (\text{rate in}) - (\text{rate out}). \]