How do you rewrite the differential equation 2ydy - ($1+\cos x$)($1+y^2$)$dx = 0$ into separable form?

Conditions

• The equation is a first-order differential equation.
• The term ($1+y^2$) is never zero for real y, so division is always valid.

Reasoning, step by step

1. Start with the original equation: 2ydy - ($1+\cos x$)($1+y^2$)$dx = 0$.
2. Move the dx term to the right side: 2ydy = ($1+\cos x$)($1+y^2$)dx.
3. Divide both sides by ($1+y^2$): (2y)/($1+y^2$)$dy = (1+\cos x)dx$.
4. Verify that the left side contains only y and dy, and the right side contains only x and dx.

Example

The video demonstrates this exact transformation on the whiteboard, showing the step-by-step algebraic manipulation to reach the separable form.

Common misconceptions

• Forgetting to divide by ($1+y^2$) and leaving the y terms mixed with the x terms.
• Assuming the equation cannot be separated because it looks complex initially.

Watch the explanation

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• Why is the presence of $y^2 a$ sign that the equation is nonlinear?