How do you convert the differential equation $dy/dx = y/(x+y^2)$ into a first-order linear equation in x?

Conditions

• The equation is a first-order differential equation.
• y is not equal to zero, allowing division by y and taking the reciprocal.

Reasoning, step by step

1. Start with the equation: $dy/dx = y/(x+y^2)$.
2. Take the reciprocal of both sides: $dx/dy = (x+y^2)/y$.
3. Split the fraction: $dx/dy = (1/y)x + y$.
4. Rearrange into standard linear form: $dx/dy - (1/y)x = y$.
5. Identify $p(y) = -1/y$ and $q(y) = y$.

Example

The video demonstrates this variable-swapping technique, showing how an equation nonlinear in y becomes linear in x.

Common misconceptions

• Trying to separate variables directly, which fails because x and y are added in the denominator.
• Forgetting to change the derivative notation from $dy/dx$ to $dx/dy$ when swapping variables.

Watch the explanation

Explore next

• How do you apply the general solution formula for a first-order linear equation in x?
• Why is the original equation considered nonlinear in y?