Mathematical Methods for Physics and Engineering : A Comprehensive Guide

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Solve

dy

dx

+

y

x

=2x^3 y^4.

If we letv=y^1 −^4 =y−^3 then

dy

dx

=−

y^4

3

dv

dx

.

Substituting this into the ODE and rearranging, we obtain

dv

dx

−

3 v

x

=− 6 x^3 ,

which is linear and may be solved by multiplying through by the integrating factor (see
subsection 14.2.4)

exp

{

− 3

∫

dx

x

}

=exp(−3lnx)=

1

x^3

.

This yields the solution

v

x^3

=− 6 x+c.

Remembering thatv=y−^3 ,weobtainy−^3 =− 6 x^4 +cx^3 .

Solution method.Rearrange the equation into the form (14.21) and make the sub-

stitutionv=y^1 −n. This leads to a linear equation inv, which can be solved by the

method of subsection 14.2.4. Then replacevbyy^1 −nto obtain the solution.

14.2.8 Miscellaneous equations

There are two further types of first-degree first-order equation that occur fairly

regularly but do not fall into any of the above categories. They may be reduced

to one of the above equations, however, by a suitable change of variable.

Firstly, we consider

dy

dx

=F(ax+by+c), (14.22)

wherea,bandcare constants, i.e.xandyonlyappear on the RHS in the particular

combinationax+by+cand not in any other combination or by themselves. This

equation can be solved by making the substitutionv=ax+by+c,inwhichcase

dv

dx

=a+b

dy

dx

=a+bF(v), (14.23)

which is separable and may be integrated directly.