Question

Let \( x(y) \) be the solution of the given differential equation \( 2y^2 \dfrac{dx}{dy} - 2xy + x^2 = 0 \). If \( x(e) = e \), then \( \dfrac{3x(e^2)}{e^2} \) equals.

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

When both \(x\) and \(y\) appear in the form \(x/y\) or \(x^2/y^2\), try the substitution \(x=vy\). It usually converts the equation into a separable differential equation.

Answer 1

The Correct Option is B

Step 1: Rewrite the differential equation in a simpler form.
Given differential equation is:
\[ 2y^2 \frac{dx}{dy}-2xy+x^2=0 \] Divide the whole equation by \(2y^2\):
\[ \frac{dx}{dy}-\frac{x}{y}+\frac{x^2}{2y^2}=0 \] \[ \frac{dx}{dy}=\frac{x}{y}-\frac{x^2}{2y^2} \] This is a homogeneous type differential equation.

Step 2: Use substitution \(x=vy\).
Let:
\[ x=vy \] Then:
\[ \frac{dx}{dy}=v+y\frac{dv}{dy} \] Substitute into the differential equation:
\[ v+y\frac{dv}{dy}=v-\frac{v^2}{2} \] Subtract \(v\) from both sides:
\[ y\frac{dv}{dy}=-\frac{v^2}{2} \] So,
\[ \frac{dv}{v^2}=-\frac{1}{2}\frac{dy}{y} \]
Step 3: Integrate both sides.
Integrating, we get:
\[ \int v^{-2}\,dv=-\frac{1}{2}\int \frac{1}{y}\,dy \] \[ -\frac{1}{v}=-\frac{1}{2}\ln y + C \] \[ \frac{1}{v}=\frac{1}{2}\ln y + C_1 \] Since \(v=\dfrac{x}{y}\), we have:
\[ \frac{1}{x/y}=\frac{y}{x}=\frac{1}{2}\ln y + C_1 \] Thus,
\[ x=\frac{y}{\frac{1}{2}\ln y + C_1} \]
Step 4: Use the initial condition \(x(e)=e\).
Given:
\[ x(e)=e \] Substitute \(y=e\):
\[ e=\frac{e}{\frac{1}{2}\ln e + C_1} \] Since \( \ln e =1 \),
\[ e=\frac{e}{\frac{1}{2}+C_1} \] So,
\[ \frac{1}{2}+C_1=1 \] \[ C_1=\frac{1}{2} \] Hence, the solution becomes:
\[ x=\frac{y}{\frac{1}{2}\ln y + \frac{1}{2}} \] \[ x=\frac{2y}{\ln y + 1} \]
Step 5: Find \(x(e^2)\).
Substitute \(y=e^2\):
\[ x(e^2)=\frac{2e^2}{\ln(e^2)+1} \] Since \( \ln(e^2)=2 \),
\[ x(e^2)=\frac{2e^2}{2+1} \] \[ x(e^2)=\frac{2e^2}{3} \]
Step 6: Evaluate the required expression.
We need to find:
\[ \frac{3x(e^2)}{e^2} \] Substitute \(x(e^2)=\dfrac{2e^2}{3}\):
\[ \frac{3x(e^2)}{e^2}=\frac{3\left(\frac{2e^2}{3}\right)}{e^2} \] \[ =\frac{2e^2}{e^2}=2 \]
Step 7: Conclusion.
Therefore, the required value is \(2\).

Final Answer: \(2\)