Question

Each DE in Problemsis a Bernoulli equation. In Problemssolve the given differential equation by using an appropriate substitution.

Short answer

Answer:

The solution of the given differential equation is.

Step-by-step solution

Define substitution method for differentiation.

Often, the first step in solving a differential equation is to use a substitution to change it into another differential equation.

In the differential equation , if can be any real number, it is called a Bernoulli’s equation. If and , Bernoulli’s equation is linear. If and , the substitution reduces Bernoulli’s equation to a linear equation.

Find the derivative of the function using chain rule.

Divide the equation on both sides by .

Multiply the equation on both sides by .

… (1)

Let , and take the derivative of with respect to .

Define the derivative of the function using the chain rule as shown below.

Substitute the values of and in equation (1).

Separate the variables and integrate the function.

Separate the variables and simplify it.

Integrate on both sides.

Substitute equation for into the above equation.

Thus, the solution of the given differential equation is.