Mathematical Methods for Physics and Engineering : A Comprehensive Guide

14.3 HIGHER-DEGREE FIRST-ORDER EQUATIONS

whereFi=Fi(x, y). We are then left with solving thenfirst-degree equations

p=Fi(x, y). Writing the solutions to these first-degree equations asGi(x, y)=0,

the general solution to (14.28) is given by the product

G 1 (x, y)G 2 (x, y)···Gn(x, y)=0. (14.29)

Solve

(x^3 +x^2 +x+1)p^2 −(3x^2 +2x+1)yp+2xy^2 =0. (14.30)

This equation may be factorised to give

[(x+1)p−y][(x^2 +1)p− 2 xy]=0.

Taking each bracket in turn we have

(x+1)

dy

dx

−y=0,

(x^2 +1)

dy

dx

− 2 xy=0,

which have the solutionsy−c(x+1) = 0 andy−c(x^2 + 1) = 0 respectively (see
section 14.2 on first-degree first-order equations). Note that the arbitrary constants in
these two solutions can be taken to be the same, since only one is required for a first-order
equation. The general solution to (14.30) is then given by

[y−c(x+1)]

[

y−c(x^2 +1)

]

=0.

Solution method.If the equation can be factorised into the form (14.28) then solve

the first-order ODEp−Fi=0for each factor and write the solution in the form

Gi(x, y)=0. The solution to the original equation is then given by the product

(14.29).

14.3.2 Equations soluble forx

Equations that can be solved forx, i.e. such that they may be written in the form

x=F(y, p), (14.31)

can be reduced to first-degree first-order equations inpby differentiating both

sides with respect toy,sothat

dx

dy

=

1

p

=

∂F

∂y

+

∂F

∂p

dp

dy

.

This results in an equation of the formG(y, p) = 0, which can be used together

with (14.31) to eliminatepand give the general solution. Note that often a singular

solution to the equation will be found at the same time (see the introduction to

this chapter).