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

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Final answer:

To solve the given differential equation, we can use the Bernoulli equation technique. The first step is to make a substitution of variables. Next, we simplify the equation, and solve it using the integrating factor method.

Step-by-step explanation:

To solve the given differential equation, we can use the Bernoulli equation technique. The Bernoulli equation is in the form x(dy/dx) - (1/x)y = xy². The first step is to make a substitution of variables. Let's substitute y = u/x, where u is a new variable. Substituting this in the equation gives us x(du/dx) + (u/x) - u/x = x(u/x)².

Next, we simplify the equation by cancelling out the x terms and expanding the square term. This gives us du/dx + u = u².

This is now a simple first-order ordinary differential equation. We can solve it using separation of variables or integrating factors. I'll use the integrating factor method. Multiply the entire equation by e^x to get e^x(du/dx) + e^xu = e^xu². This can be rewritten as d/dx(e^xu) = e^xu².

Integrating both sides, we get e^xu = -1/(u+c), where c is the constant of integration.

Simplifying and solving for u, we have u = -1/(e^x(u+c)). However, we substituted y = u/x earlier, so substituting back gives us y = -1/(xe^x(y+c)). This is the solution to the given differential equation.

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