Find the general solution of the following differential equation:
\(x^{2}\dfrac{dy}{dx} = x^{2} + xy + y^{2}\)
Find the general solution of the following differential equation:
\(x^{2}\dfrac{dy}{dx} = x^{2} + xy + y^{2}\)
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Solution and Explanation
Divide both sides by \(x^2\) (with \(x \neq 0\)).
\[ \frac{dy}{dx} = 1 + \frac{y}{x} + \left(\frac{y}{x}\right)^2 \]
The right side depends only on \(\dfrac{y}{x}\), so the equation is homogeneous.
Step 2: Substitute y = vx:
Put \(y = vx\), so \(\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}\).
The equation becomes the following.
\[ v + x\frac{dv}{dx} = 1 + v + v^2 \]
The term \(v\) cancels on both sides.
\[ x\frac{dv}{dx} = 1 + v^2 \]
Step 3: Separate the variables:
Keep \(v\) on one side and \(x\) on the other.
\[ \frac{dv}{1+v^2} = \frac{dx}{x} \]
Step 4: Integrate both sides:
The left side is a standard integral that gives an inverse tangent.
\[ \tan^{-1} v = \log|x| + C \]
Step 5: Replace v by y/x:
Put \(v = \dfrac{y}{x}\) back and solve for \(y\).
\[ \tan^{-1}\left(\frac{y}{x}\right) = \log|x| + C \quad \Rightarrow \quad y = x\tan\left(\log|x| + C\right) \]
Final Answer:
The general solution, with C as the arbitrary constant, is below. \[ \boxed{y = x\tan\left(\log|x| + C\right)} \]
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