The solution of the differential equation $x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+4y=0$ is equal to:

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To solve the differential equation $x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+4y=0$, we recognize it as a Cauchy-Euler equation, which is a type of linear differential equation with variable coefficients. The general form of a Cauchy-Euler equation is:

$$ a_n x^n \frac{d^n y}{dx^n} + a_{n-1} x^{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1 x \frac{dy}{dx} + a_0 y = 0. $$

In this case, the equation is:

$$ x^2 \frac{d^2 y}{dx^2} - 3x \frac{dy}{dx} + 4y = 0. $$

To solve this, we assume a solution of the form $ y = x^m $. Substituting $ y = x^m $ into the differential equation, we find:

1. The first derivative: $\frac{dy}{dx} = mx^{m-1}$.
2. The second derivative: $\frac{d^2y}{dx^2} = m(m-1)x^{m-2}$.

Substituting these into the differential equation gives:

$$ x^2(m(m-1)x^{m-2}) - 3x(mx^{m-1}) + 4x^m = 0. $$

Simplifying, we have:

$$ m(m-1)x^m - 3mx^m + 4x^m = 0. $$

This simplifies to:

$$ (m(m-1) - 3m + 4)x^m = 0. $$

The characteristic equation is:

$$ m^2 - 4m + 4 = 0. $$

Factoring, we find:

$$ (m-2)^2 = 0. $$

This indicates a repeated root $ m = 2 $. For a repeated root in a Cauchy-Euler equation, the general solution is:

$$ y = c_1 x^m + c_2 x^m \ln x. $$

Substituting $ m = 2 $, the solution becomes:

$$ y = c_1 x^2 + c_2 x^2 \ln x. $$

This matches Option 4: $ y = (c_1 + c_2 \ln x)x^2 $.

Now, let's explain why the other options are incorrect:

1. **Option 1: $ y = (c_1 + c_2 x)e^{2x} $** - This form suggests an exponential solution, which is not applicable to a Cauchy-Euler equation with polynomial solutions.

2. **Option 2: $ y = (c_1 + c_2 x)e^{x} $** - Similar to Option 1, this implies an exponential solution, which does not fit the polynomial nature of the Cauchy-Euler equation.

3. **Option 3: $ y = (c_1 + c_2 x)\ln x $** - This form suggests a solution involving $\ln x$ without the $x^2$ term, which does not satisfy the characteristic equation derived from the given differential equation.

Thus, Option 4 is the correct solution, as it accurately reflects the solution form for a Cauchy-Euler equation with a repeated root.