Find the general solution of the differential equation :
\(y^2\,dx + (x^2 - xy + y^2)\,dy = 0\)
Find the general solution of the differential equation :
\(y^2\,dx + (x^2 - xy + y^2)\,dy = 0\)
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Solution and Explanation
The coefficients of \(dx\) and \(dy\) are both homogeneous of degree 2. So the equation is homogeneous and we can use the substitution \(x = vy\).
Step 2: Write the equation as dx/dy.
\[ \frac{dx}{dy} = -\frac{x^2 - xy + y^2}{y^2} \]
Step 3: Substitute x = vy.
Then \(\dfrac{dx}{dy} = v + y\dfrac{dv}{dy}\) and the right side becomes \(-(v^2 - v + 1)\).
\[ v + y\frac{dv}{dy} = -v^2 + v - 1 \]
\[ y\frac{dv}{dy} = -(v^2 + 1) \]
Step 4: Separate the variables and integrate.
\[ \frac{dv}{v^2 + 1} = -\frac{dy}{y} \]
\[ \tan^{-1} v = -\log|y| + C \]
Step 5: Put back v = x/y.
\[ \tan^{-1}\left(\frac{x}{y}\right) + \log|y| = C \]
Final Answer:
The general solution is given below, where \(C\) is an arbitrary constant. \[ \boxed{\tan^{-1}\left(\frac{x}{y}\right) + \log|y| = C} \]
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