Question:
The general solution of the differential equation: \( x^2 dy + y^2 dx = 0 \) is
The general solution of the differential equation: \( x^2 dy + y^2 dx = 0 \) is
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When separating variables, always move the differentials \( dx \) and \( dy \) to the numerators first.
Constant terms can be rewritten (e.g., \( -C \) as \( k \)) to match the format of given options.
Constant terms can be rewritten (e.g., \( -C \) as \( k \)) to match the format of given options.
Updated On: Sep 10, 2026
- \( x^3 + y^3 = k \)
- \( \frac{1}{x} - \frac{1}{y} = k \)
- \( \frac{1}{y} + \frac{1}{x} = k \)
- \( \log y^2 + \log x^2 = k \)
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The Correct Option is C
Solution and Explanation
Concept:
• Variable Separable Method: A technique to solve differential equations by moving all terms involving \( y \) to one side and all terms involving \( x \) to the other.
• Integration of power functions: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \).
Step 1: Separate the variables
The given equation is \( x^2 dy + y^2 dx = 0 \).
Rearranging:
\[ x^2 dy = -y^2 dx \]
Divide both sides by \( x^2 y^2 \) (assuming \( x, y \neq 0 \)):
\[ \frac{dy}{y^2} = -\frac{dx}{x^2} \]
\[ y^{-2} dy = -x^{-2} dx \]
Step 2: Integrate both sides
\[ \int y^{-2} dy = \int -x^{-2} dx \]
Applying the integration formula:
\[ \frac{y^{-1}}{-1} = -\left( \frac{x^{-1}}{-1} \right) + C \]
\[ -\frac{1}{y} = \frac{1}{x} + C \]
Step 3: Simplify to the standard form
Multiply the equation by \( -1 \):
\[ \frac{1}{y} = -\frac{1}{x} - C \]
\[ \frac{1}{y} + \frac{1}{x} = -C \]
Let \( k = -C \) be the arbitrary constant:
\[ \frac{1}{y} + \frac{1}{x} = k \]
This matches option (C).
• Variable Separable Method: A technique to solve differential equations by moving all terms involving \( y \) to one side and all terms involving \( x \) to the other.
• Integration of power functions: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \).
Step 1: Separate the variables
The given equation is \( x^2 dy + y^2 dx = 0 \).
Rearranging:
\[ x^2 dy = -y^2 dx \]
Divide both sides by \( x^2 y^2 \) (assuming \( x, y \neq 0 \)):
\[ \frac{dy}{y^2} = -\frac{dx}{x^2} \]
\[ y^{-2} dy = -x^{-2} dx \]
Step 2: Integrate both sides
\[ \int y^{-2} dy = \int -x^{-2} dx \]
Applying the integration formula:
\[ \frac{y^{-1}}{-1} = -\left( \frac{x^{-1}}{-1} \right) + C \]
\[ -\frac{1}{y} = \frac{1}{x} + C \]
Step 3: Simplify to the standard form
Multiply the equation by \( -1 \):
\[ \frac{1}{y} = -\frac{1}{x} - C \]
\[ \frac{1}{y} + \frac{1}{x} = -C \]
Let \( k = -C \) be the arbitrary constant:
\[ \frac{1}{y} + \frac{1}{x} = k \]
This matches option (C).
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