The differential equation of which `y=ax^(2)+bx` is the general solution, a, b being arbitrary constants is `x^(2)(d^(2)y)/(dx^(2))-2x(dy)/(dx) +2y=0`.

Answer

To solve the problem, we need to verify if the function \( y = ax^2 + bx \) is a solution to the differential equation given by

\[ x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0 \]

where \( a \) and \( b \) are arbitrary constants. We will do this by differentiating \( y \) and substituting the derivatives into the differential equation.

### Step 1: Differentiate \( y \)

Given:
\[ y = ax^2 + bx \]

First, we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = \frac{d}{dx}(ax^2 + bx) = 2ax + b \]

### Step 2: Differentiate \( \frac{dy}{dx} \)

Next, we differentiate \( \frac{dy}{dx} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(2ax + b) = 2a \]

### Step 3: Substitute into the differential equation

Now we substitute \( y \), \( \frac{dy}{dx} \), and \( \frac{d^2y}{dx^2} \) into the differential equation:

\[ x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0 \]

Substituting the values we found:

\[ x^2(2a) - 2x(2ax + b) + 2(ax^2 + bx) = 0 \]

### Step 4: Simplify the equation

Now we simplify the left-hand side:

\[ 2ax^2 - 2(2ax^2 + bx) + 2(ax^2 + bx) = 0 \]

Expanding this gives:

\[ 2ax^2 - 4ax^2 - 2bx + 2ax^2 + 2bx = 0 \]

Combining like terms:

\[ (2a - 4a + 2a)x^2 + (-2b + 2b) = 0 \]

This simplifies to:

\[ 0 = 0 \]

### Conclusion

Since the left-hand side equals zero, the function \( y = ax^2 + bx \) satisfies the differential equation. Therefore, the statement is true.