First order higher degree (nonlinear)

General form #

$$ \begin{align*} f(x,y,y’)=0\\ or\\ f(x,y,p)=0;\\ p=y’\\ \\ p^n+a_1p^{n-1}+a_2p^{n-2}+…+a_{n-1}p+a_{n}=0;\\ a_k(x,y) \end{align*} $$

Methods #

Method 1: Equations solvable for p #

$$ \begin{align*} p^n+a_1p^{n-1}+a_2p^{n-2}+…+a_{n-1}p+a_n=0\\ (p-f_1(x,y))(p-f_2(x,y))…(p-f_n(x,y))=0\\

Solving\\ n\\ equations:\\ y’-f_k(x,y)=0: F_k(x,y,c)=0\\ Final\\ solution:\\ F_1(x,y,c)F_2(x,y,c)…F_n(x,y,c)=0 \end{align*} $$

Method 2: Equations solvable for x #

$$ \begin{align*} Solving\\ f(x,y,p)=0\\ for\\ x, x=F(y,p)\\ \dd{x}{y}=\pd{F}{y}+\pd{F}{p}\dd{p}{y}\\ 0=ϕ_1(y,p,\dd{p}{y})ϕ_2(y,p); first\\ order,degree\\ \text{we ignore }ϕ_2\\ \text{Sub solution: } \Phi(y,p,c)=0\\ \text{Final solution: Eliminating p from } x=F(y,p),\Phi \end{align*} $$

Method 3: Equations solvable for y #

Same as

Clairaut’s Equation #

Form #

$$ y=xp+\psi (p) $$

Solution #

$$ \begin{align*} \dd{y}{x}= p+&x\dd{p}{x}+\psi’(p)\dd{p}{x}\\ 0= &x\dd{p}{x}+\psi’(p)\dd{p}{x}\\ (x+\psi’(p))\dd{p}{x}=0 \end{align*} $$

General Sol #

$$ \begin{align*} \dd{p}{x}=0\\ Subbing\\ p=c\\ in\\ main\\ y=xc+\psi (c) \end{align*} $$

Singular solution #

$$ \begin{align*} Eliminating\\ p\\ from:\\ (x+\psi ‘(x))=0\\ y=xp+\psi (p) \end{align*} $$

Lagrange’s Equation #

Form #

$$ y=x\phi (p)+\Phi (p) $$