Differential Equations Question - None

Question

The family of curves represented by $\frac{dy}{dx} = \frac{x + y}{x}$ is

Accepted Answer

Step 1: Rewrite the differential equation: $\frac{dy}{dx} = 1 + \frac{y}{x}$

Step 2: This is a linear differential equation of the form $\frac{dy}{dx} - \frac{y}{x} = 1$

Step 3: The integrating factor is $e^{\int -\frac{1}{x}dx} = e^{-\ln x} = \frac{1}{x}$

Step 4: Multiply both sides by the integrating factor: $\frac{1}{x}\frac{dy}{dx} - \frac{y}{x^2} = \frac{1}{x}$

Step 5: The left side is $\frac{d}{dx}\left(\frac{y}{x}\right) = \frac{1}{x}$

Step 6: Integrate both sides: $\frac{y}{x} = \ln(x) + C$

Step 7: Therefore $y = x\ln(x) + Cx$

∴ Answer is A.

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