A differential equation is given as The solution of the differential equation in terms of arbitrary constants C1 and C2 is bhjvhhgghvhbjj

The correct option is:
$$y=C_1{x}^2+C_{2}x+2$$

Analysis of the Question:
From the provided images, the differential equation to solve is:
$$x^2\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+2y=4$$

This equation is a second-order Cauchy-Euler differential equation (also known as a homogeneous linear differential equation).

Step 1: Substitution to constant coefficient form
To solve this, we introduce a new independent variable $z$ using the substitution:
$$x=e^z\text{or}z=\ln \left(x\right)$$

Using the chain rule, we can relate the derivatives with respect to $x$ to derivatives with respect to $z$. Let $D=\frac{d}{dz}$:
$$x\frac{dy}{dx}=Dy$$
And for the second derivative:
$$x^2\frac{d^{2}y}{d{x}^2}=D(D-1)y$$

Substituting these expressions into our original differential equation gives:
$$\left[D\right(D-1)-2D+2]y=4$$
Expanding and simplifying the operator:
$$(D^2-D-2D+2)y=4$$
$$(D^2-3D+2)y=4$$

Step 2: Finding the Complementary Function (CF)
To find the complementary function $y_c$, we solve the homogeneous equation:
$$(D^2-3D+2)y=0$$
The corresponding auxiliary algebraic equation is:
$$m^2-3m+2=0$$
Factoring the quadratic expression:
$$(m-1)(m-2)=0$$
Which yields the real and distinct roots:
$$m_1=1,m_2=2$$

Thus, the solution of the homogeneous equation in terms of $z$ is:
$$y_c=C_1{e}^{2z}+C_2{e}^z$$
Substituting $e^z=x$ back into the equation:
$$y_c=C_1{x}^2+C_{2}x$$

Step 3: Finding the Particular Integral (PI)
The particular integral $y_p$ is calculated for the non-homogeneous term:
$$y_p=\frac{1}{D^2-3D+2}\left(4\right)$$
We can write the constant $4$ as $4e^{0z}$. According to the rules for finding the PI of an exponential term $e^{az}$, we substitute $D=a=0$:
$$y_p=\frac{4}{0^2-3\left(0\right)+2}=\frac{4}{2}=2$$

Step 4: Writing the General Solution
The complete general solution is the sum of the complementary function and the particular integral:
$$y=y_c+y_p$$
$$y=C_1{x}^2+C_{2}x+2$$