Reduction of Order

Solved Problems

Solution.

This example relates to the Case $$ Consider the function $$ Then $$ Consequently,

Integrating, we find the function $$

Given that $$ we integrate one more equation of the $$st order:

The latter formula gives the general solution of the original differential equation.

Solution.

This is an equation of type $$ where the right-hand side depends only on the variable $$ We introduce the parameter $$ Then the equation can be written as

We obtain the equation of the $$st order for the function $$ with separable variables. Integrating gives:

where $$ is a constant of integration.

Taking the square root of both sides, we find the function $$

Now recall that $$ and solve another equation of the $$st order:

Separate the variables and integrate:

To calculate the integral on the left-hand side, make the replacement:

Then the left-hand integral is equal to

As a result, we obtain the following algebraic equation:

where $$ are constants of integration.

The last expression is the general solution of the differential equation in implicit form.

Solution.

This equation does not contain the function $$ and the independent variable $$ (Case $$). Therefore, we set $$ Then this equation takes the form

The resulting first-order equation for the function $$ is a separable equation and can be easily integrated:

Replacing $$ by $$ we obtain

Integrating again, we find the general solution of the original differential equation:

Solution.

This equation does not explicitly include the variable $$, i.e. it corresponds to the type $$ in our classification. We introduce the new variable $$ The original equation is transformed into the first order equation:

which is solved by separation of variables:

Integrating the resulting equation once more yields the function $$

To compute the last integral we make the substitution: $$ As a result, we have

Returning to the variable $$ we finally obtain

Solution.

This equation does not explicitly contain the independent variable $$ that is refers to the Case $$ Let $$ Then the equation can be written as

Separate variables and integrate:

Integrating again, we obtain the final solution in implicit form:

where $$ are constants of integration.

Solution.

The equation satisfies the condition of homogeneity. Therefore, we make the following change of variable: $$ The derivatives will be equal

Then the differential equation becomes:

It is easy to find the function $$

The original function $$ is defined by the formula

The calculations give the following answer:

Note that in addition to the general solution, the differential equation also contains the singular solution $$

Solution.

You may notice that the left side of the equation is the derivative of $$ Therefore, denoting $$ we obtain the following differential equation:

The last equation is easily solved by separation of variables:

Now we integrate one more equation for $$

where $$ are arbitrary constants.