HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS
15.3 General ordinary differential equationsIn this section, we discuss miscellaneous methods for simplifying general ODEs.
These methods are applicable to both linear and non-linear equations and in
some cases may lead to a solution. More often than not, however, finding a
closed-form solution to a general non-linear ODE proves impossible.
15.3.1 Dependent variable absentIf an ODE does not contain the dependent variableyexplicitly, but only its
derivatives, then the change of variablep=dy/dxleads to an equation of one
order lower.
Solve
d^2 y
dx^2+2
dy
dx=4x (15.76)This is transformed by the substitutionp=dy/dxto the first-order equation
dp
dx+2p=4x. (15.77)The solution to (15.77) is then found by the method of subsection 14.2.4 and reads
p=dy
dx=ae−^2 x+2x− 1 ,whereais a constant. Thus by direct integration the solution to the original equation,
(15.76), is
y(x)=c 1 e−^2 x+x^2 −x+c 2 .An extension to the above method is appropriate if an ODE contains onlyderivatives ofythat are of ordermand greater. Then the substitutionp=dmy/dxm
reduces the order of the ODE bym.
Solution method.If the ODE contains only derivatives ofythat are of ordermand
greater then the substitutionp=dmy/dxmreduces the order of the equation bym.
15.3.2 Independent variable absentIf an ODE does not contain the independent variablexexplicitly, except ind/dx,
d^2 /dx^2 etc., then as in the previous subsection we make the substitutionp=dy/dx