Substitutions

Homogeneous Equations

• If $f(x,y)$ in $\frac{dy}{dx}=f(x,y)$ can be expressed in terms of $\frac{y}{x}$ alone
• Used in the case that all variables have the same power
  • There needs to be at least one dependent variable

Steps

1. Make the substitution $v=\frac{y}{x}$, $\frac{dy}{dx}=v+x\frac{dv}{dx}$
   • Sometimes it is better to use $dy=vdx+xdv$
2. Solve as a Separable DEQs

Bernoulli Equations

First order equation in the form

$$\frac{dy}{dx}+P(x)y=Q(x)y^n$$

where $P(x)$ and $Q(x)$ are continuous on an interval $(a,b)$ and $n$ is a real number

All linear equations are Bernoulli where the $n=0$ (but you would just want to solve them as linear)

Steps

1. Divide by $y^n$
2. Substitute $v=y^{1-n}$, $\frac{dv}{dx}=(n-1)y^{-n}\frac{dy}{dx}$
3. Solve as a First Order Linear DEQs

Equation of Linear Combination

When $f$ in $\frac{dy}{dx}=f(x,y)$ can be expressed as a function of $ax+by+c$

Steps

1. Substitute $z=ax+by+c$
2. Solve as a Separable DEQs