1.  Equations of the form

y(n) = f (x)

are easily solved by direct integration.

Really,

,

where  f1 (x)  is a primitive of  f (x).

Then

,

and so on.

2.  Let us consider equations

,

which do not contain the function itself and its first (k – 1) derivatives.

Applying the substitution

,

we obtain the equation of (n-k)-th order,

,

the solving of which is a simpler problem.

Let us assume that

is the general solution of the obtained equation.

Then

is a directly integrable differential equation considered above.

3.  If an equation does not contain (in explicit form) the variable x,

,

then by making use of substitution

we have

and so on.

Therefore, the original equation of the n-th order is reduced to an equation of the order (n – 1). If

is the general solution of the obtained equation, then

which is a differential equation with separated variables.

Indefinite Integrals Definite Integrals Differential Equations