Techniques of solving order 1 ODEs
Separable form
If a differential equation is of the form $\frac{dy}{dx} = \frac{P(x)}{Q(y)}$ (or equivalently $y’=P(x)Q(y)$), then we solve the equation by \(\int P(x)dx = \int Q(y)dy.\)
Integral factor (non separable form)
If a differential equation is of the form $y’+P(x)y=Q(x)$ (note that the leading coefficient is 1), then we will find its integral factor \(I=e^{\int P(x)dx}\) and the solution is \(y=\int IQ(x)dx.\)
Direction Field
Direction Field of the logistic model $P’ = k P(1-\frac{1}{P})$
Direction Field of the logistic model $y’ = \frac{y}{x}$
Direction Field of the logistic model $y’ = -\frac{x}{y}$
Direction Field of the logistic model $y’ = \left(\frac{y}{x}\right)^2$
Direction Field of the logistic model $y’ = -\left(\frac{x}{y}\right)^2$
Orthogonal trajectories
Given a family of curves ${C_k}$ where $k$ is a parameter that can be expressed by $x$ and $y$, an orthogonal trajectory of the family is a curve $C$ such that every intersection of $C$ and $C_i$ are perpendicular. To find $C$, we follow the following steps:
Step 1 Find $\frac{dy}{dx}=f(x,y)$ of $C_k$
Step 2 Replace $k$ in $f(x,y)$ in terms of $x$ and $y$.
Step 3 Let $\frac{dy}{dx} = -\frac{1}{f(x,y)}$
Step 4 Solve the last ODE.