How do you separate variables in the differential equation ($1+e^x$)dy = ye^x dx?

Conditions

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

Reasoning, step by step

1. Start with the equation: ($1+e^x$)dy = ye^x dx.
2. Divide by ($1+e^x$): dy = (ye^$x/(1+e^x)$)dx.
3. Divide by y: $dy/y = (e^x/(1+e^x))dx$.
4. Verify that the left side contains only y and dy, and the right side contains only x and dx.

Example

The instructor demonstrates this two-step division process on the board to achieve the separated form.

Common misconceptions

• Dividing by y before dividing by ($1+e^x$), which complicates the algebra unnecessarily.
• Forgetting to check if $y=0$ is a valid solution that might be lost during division.

Watch the explanation

Explore next

• How do you integrate $dy/y = (e^x/(1+e^x))dx$?
• Why is this equation also considered a first-order linear equation?