Question

Solve the given differential equation by using an appropriate substitution. The DE is a Bernoulli equation. dx/dy = y(xy² − 1)

Answer 1

To solve the given Bernoulli equation dx/dy = y(xy² − 1), we can use the substitution u = y². This will transform the equation into a linear differential equation. Follow the step-by-step outlined in the answer to solve the equation.

Step-by-step explanation:

To solve the given Bernoulli equation dx/dy = y(xy² − 1), we can use the substitution u = y². This will transform the equation into a linear differential equation. Here are the steps:

1. Let u = y²
2. Find the derivative du/dy: du/dy = 2y
3. Substitute the value of u and du/dy into the original equation: dx/dy = (u/2)(x - 1)
4. Rewrite the equation as dx/dy - (x - 1)u/2 = 0
5. This is now a first-order linear differential equation which can be solved using an integrating factor or by using an integrating factor formula.
6. After solving the linear equation, substitute back into the original substitution u = y² to find the value of y.