Techniques

Formula for Linear Separable First order ODE

1. Write in standard form

$$y^{\prime }+p(x)y=q(x)$$

2. Determine integrating factor

$$I(x)=e^{\int p(x)dx}$$

3. General solution

$$y(x)=\frac{1}{I(x)}\left(\int I(x)Q(x)dx+C\right)$$

Only add the integrating factor here, at the end

Bernoulli Equations

In the form:

$$y^{\prime }+p(x)y=f(x)y^r$$

Where r is any real number other than 0 or 1.

1. Divide by $y^r$

$$y^{\prime }{y}^{-r}+p(x)y^{1-r}=f(x)$$

2. Substitute $v=y^{1-r}$ and it’s derivative $\frac{dv}{dx}=(1-r)y^{-r}\frac{dy}{dx}$ Following U-Sub rules.. $\to \frac{dy}{dx}=\frac{dv}{dx}(1-r)y^r$

$$\frac{dv}{dx}(1-r)y^r{y}^{-r}+p(x)v=f(x)$$

$y^r$ and $y^{-r}$ cancel..

$$\frac{dv}{dx}(1-r)+p(x)v=f(x)$$

3. This is now linear, first order, and separable. Solve as such
4. Back substitute, and solve for y