Question:
The solution of the differential equation $x\frac{dy}{dx} + y = 0$ passing through the point (1,1) is $y =$
The solution of the differential equation $x\frac{dy}{dx} + y = 0$ passing through the point (1,1) is $y =$
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$x\frac{dy}{dx} + y = 0$ is just the expansion of $\frac{d}{dx}(xy) = 0$. This means $xy = c$ immediately!
Updated On: May 5, 2026
- $x^2$
- $x^{-1}$
- $x^{-2}$
- $x$
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Solve the first-order differential equation using the separation of variables.
Step 2: Meaning
Rearrange the terms: $x \frac{dy}{dx} = -y \implies \frac{dy}{y} = -\frac{dx}{x}$.
Step 3: Analysis
Integrate both sides: $\int \frac{1}{y} dy = -\int \frac{1}{x} dx \implies \log y = -\log x + \log c \implies \log y = \log(\frac{c}{x})$. Thus, $y = \frac{c}{x}$.
Step 4: Conclusion
Pass through (1,1): $1 = \frac{c}{1} \implies c = 1$. So, $y = \frac{1}{x} = x^{-1}$.
Final Answer: (B)
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