F. First Order Bernoulli Differential Equation

A first order Bernoulli differential equation follows the form

The solution to this equation may be obtain by transforming the original equation into a linear differential equation by the substitution of variable:

Substituting to the original Bernoulli Differential Equation, we obtain:

Multiplying both sides of the equation by (1 - n)y-n we obtain the the FOLDE:

The solution to the Bernoulli Differential Equation is obtained by the equation:

where

Example: Determine the solution of

Solution:

Transform the given DE to a form

or

From the previous equation we can specify the following:

thus

Now, the determination of the solution rests on the two formulas that were given on the previous page:

Get the integrating factor by using the formula

Get the solution by using the formula

The solution therefore is:

SPECIAL HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS

The nth order ODE Reducible to First Order

Test for Reducibility to first order:

Illustration: Test if the 4th order ODE

is reducible to first

order.

The first thing to do is to combine all the derivatives on the left-hand side of the equation:

Just by performing visual inspection, the given DE fails the test for reducibility due to the following reasons:

Since the order n of the DE is 4, we should be able to factor

out a , which in this case is . This step is

impossible to do since the lowest derivative is .

The right-hand side of the equation is not a function of x alone.

Illustration: Test if the third order ODE

is reducible to first order.

Combining all the derivatives on the left-hand side of the equation:

Since the order n is 3, we should be able to factor out a .

Doing so, we will get:

Note that the quantity inside the square bracket symbol satisfies

the form and the right-hand side of the equation

is a function of x alone, the given DE therefore is reducible to first order.

Determination of the solution of an nth order ODE

Two major steps are needed to obtain the solution of this kind of higher ODE:

Step 1: Reduce the nth order DE to first order by performing (n-1) successive integrations.

Step 2: Solve the reduced DE as a first-order order differential equation. Take note that the complete solution will be obtained at this stage. For the solution to be complete it should have n arbitrary constants.

Example: Determine the complete solution of

.

This DE has already passed the test for reducibility, which is shown on the previous page. Same as what was done before, we should put all the derivatives together on the left-hand side of the equation.

Step 1: The order n of the given DE is 3 hence, 2 successive integrations are needed to reduce the order to 1.

First integration, factor out a :

Performing separation of variables and taking the integral of both sides:

Evaluating the integral will yield the equation:

Second integration. Again, factor out a , but this time, we will

apply this to the equation that we left during the first integration,

which is .

Performing separation of variables and taking the integral of both sides:

Evaluating the integral will yield the first order differential equation:

G. Step 2: Solve the reduced DE as a first-order order differential equation. From Step 1, we

obtained . This

first order DE is linear in y since it follows the

form where

and .

The complete solution is therefore:

We can simplify the solution by defining a new as

As a final remark, always check if the number of arbitrary constants is equal to the order of the original differential equation. In the problem that was just solved, the requirement for the presence of 3 arbitrary constants in the complete solution was satisfied since we were given a third order DE.

1.0 Second Order ODE (Dependent Variable y is absent)For this kind of differential equation, the standard form is

.

The following transformations are needed to reduce this kind of DE to first order:

Let and

Step 1: Reduce to

and solve the latter as a first

order DE. The solution that should be obtained at this stage should be in terms of p and x.

Step 2: Substitute to the solution obtained in Step 1

and re-solve as another first order DE. The solution that should be obtained at this stage should be in terms of x and y.

Also, for the solution to be complete, it should contain two arbitrary constants.

Example: Determine the complete solution of

We can rewrite this given second order DE as

. To check if this DE is under the

category of second order DE where the dependent variable y is absent, two conditions

should be satisfied, namely:

the right-hand side is zero the left-hand side should contain only the permissible terms

which are and .

Since the two conditions were satisfied, we can solve the given DE using the following steps:

Step 1: Reduce the given DE, which is

, to first order by

substituting and . Doing

what has just been suggested will enable us to treat the given differential equation as a first order DE, which is

. The latter can be solved

using either the techniques of homogeneous or linear first order DEs.

The expected solution is .

Step 2: Substitute to the solution

which was obtained in Step 1. After the substitution, this will give us another first order DE to solve

. Note that the foregoing is a

variables separable DE that has a solution of:

The equation that was obtained from Step 2, namely

is the complete solution to the given

second order DE since it has two arbitrary constants and it already contains the appropriate variables which are x and y.

2.0 Second Order ODE (Independent Variable x is absent)

For this kind of differential equation, the standard form is

.

The following transformations are needed to reduce this kind of DE to first order:

Let and

Step 1: Reduce to

and solve the latter as a

first order

DE. The solution that should be obtained at this stage should be in terms of p and y.

Step 2: Substitute to the solution obtained in Step 1

and re-solve as another first order

DE. The solution that should be obtained at this stage should be in terms of x and y. Also, for the solution to be complete, it should contain two arbitrary constants.

Example: Determine the complete solution of

We can rewrite this given second order DE as

. To check if this DE is

under the category of second order DE where the independent variable x is absent, two

conditions should be satisfied, namely:

the right-hand side is zero the left-hand side should contain only the permissible terms

which are and .

Since the two conditions were satisfied, we can solve the given DE using the following steps:

Step 1: Reduce the given DE, which is

to first order by

substituting and . Doing

what has just been suggested will enable us to treat the given differential equation as a first order DE, which is

. The latter can be solved

using separation of variables.

The expected solution is .

Step 2: Substitute to the solution

which was obtained in Step 1. After the substitution, this will give us another first order DE to solve

. Note that the foregoing is a

variables separable DE that has a solution of:

The equation that was obtained from Step 2, namely

is the

complete solution of the given second order DE since it has two arbitrary constants and it

already contains the appropriate variables which are x and y.