Algebra

dy/dx + y = xy^2

Step-by-step solution with explanation

Final Answer

(general solution), where is an arbitrary constant. The trivial solution is also valid.

Step-by-step solution

Identify This as a Bernoulli Equation

This equation has the form with . That makes it a Bernoulli equation, which we solve with a special substitution.

Divide Both Sides by y²

Dividing every term by sets up the substitution. This moves the nonlinear term to the right and isolates on the left.

Substitute v = y⁻¹ to Linearize

Let . Differentiating gives , so . Substituting turns the equation into a linear ODE in .

Rewrite as Linear ODE in v

Replace with and with . Then multiply through by to get the standard linear form .

Find the Integrating Factor

For , here . The integrating factor is . Multiplying both sides by this factor makes the left side a perfect derivative.

Multiply Through and Integrate Both Sides

The left side collapses to . For the right side, use integration by parts: .

Solve for v, Then Back-Substitute

Divide both sides by to get . Now replace with to return to the original variable.

Write the General Solution for y

Substitute back in. Also note that is a trivial solution (it satisfies the original equation), but it is already captured in the limit as , so the general solution covers all cases.

Understanding this problem

Learning Insight

Bernoulli equations are nonlinear, but the substitution always converts them into a standard linear ODE. This works because the substitution 'absorbs' the nonlinear power of , revealing a hidden linear structure underneath.

Quick Tip

Spot a Bernoulli equation fast: if your ODE looks like with , immediately divide by and substitute . Here , so .

Common Mistake

Students forget to flip the sign when substituting , writing instead of . This leads to the wrong sign on in the linear ODE and an incorrect integrating factor.