How do you convert the differential equation (y - x sin x)dx + xdy = 0 into the standard form of a first-order linear equation?

Conditions

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

Reasoning, step by step

1. Start with the original equation: (y - x sin x)dx + xdy = 0.
2. Move the dx term to the right side: xdy = -(y - x sin x)dx.
3. Divide by dx and x: $dy/dx = -(y - x \sin x)/x$.
4. Split the fraction: $dy/dx = -y/x + \sin x$.
5. Rearrange into standard form: y' + ($1/x$)$y = \sin x$.

Example

The instructor performs this step-by-step algebraic manipulation on the board to prepare the equation for applying the linear solution formula.

Common misconceptions

• Failing to distribute the negative sign correctly when moving terms.
• Forgetting to divide by x, leaving the equation in a non-standard form.

Watch the explanation

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