If `x^(2)(dy)/(dx)=(y(x+y))/(2)`, then ` (y-x)^(2) =k x^(2)y`

Answer

To solve the differential equation given by

\[ x^2 \frac{dy}{dx} = \frac{y(x+y)}{2}, \]

we will follow these steps:

### Step 1: Rearranging the Equation

First, we can rearrange the equation to isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \frac{y(x+y)}{2x^2}. \]

### Step 2: Substituting \(y = vx\)

Next, we will use the substitution \(y = vx\), where \(v\) is a function of \(x\). Then, we differentiate \(y\) with respect to \(x\):

\[ \frac{dy}{dx} = v + x \frac{dv}{dx}. \]

### Step 3: Substitute into the Equation

Substituting \(y = vx\) and \(\frac{dy}{dx}\) into the rearranged equation gives:

\[ v + x \frac{dv}{dx} = \frac{vx(x + vx)}{2x^2}. \]

This simplifies to:

\[ v + x \frac{dv}{dx} = \frac{v(x + vx)}{2x}. \]

### Step 4: Simplifying the Right Side

Now, simplifying the right-hand side:

\[ \frac{v(x + vx)}{2x} = \frac{vx + vvx}{2x} = \frac{v}{2}(1 + v). \]

### Step 5: Rearranging the Equation

Now we have:

\[ v + x \frac{dv}{dx} = \frac{v}{2}(1 + v). \]

Rearranging gives:

\[ x \frac{dv}{dx} = \frac{v}{2}(1 + v) - v = v\left(\frac{1 + v}{2} - 1\right) = v\left(\frac{v - 1}{2}\right). \]

### Step 6: Separating Variables

Now we can separate the variables:

\[ \frac{2}{v(v-1)} dv = \frac{dx}{x}. \]

### Step 7: Integrating Both Sides

Integrating both sides:

\[ \int \frac{2}{v(v-1)} dv = \int \frac{dx}{x}. \]

Using partial fractions on the left side:

\[ \frac{2}{v(v-1)} = \frac{2}{v} + \frac{2}{1-v}. \]

Integrating gives:

\[ 2 \ln |v| - 2 \ln |1-v| = \ln |x| + C. \]

### Step 8: Simplifying the Result

This simplifies to:

\[ \ln \left|\frac{v^2}{1-v}\right| = \ln |x| + C. \]

Exponentiating both sides leads to:

\[ \frac{v^2}{1-v} = kx, \]

where \(k = e^C\).

### Step 9: Substituting Back for \(y\)

Recall \(v = \frac{y}{x}\), so substituting back gives:

\[ \frac{\left(\frac{y}{x}\right)^2}{1 - \frac{y}{x}} = kx. \]

Multiplying through by \(x^2(1 - \frac{y}{x})\) leads to:

\[ y^2 = kx^2y - kx^3. \]

### Step 10: Rearranging to the Required Form

Rearranging gives:

\[ (y - x)^2 = kx^2y. \]

Thus, we have shown that the solution to the differential equation is indeed of the form:

\[ (y - x)^2 = kx^2y. \]

### Conclusion

Therefore, the statement is true.

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