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

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

Step-by-step Explanation:

1. Identify the type of differential equation:
From the first image, the given differential equation is:
$x^2\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+2y=4$
This is a second-order Cauchy-Euler (homogeneous linear) differential equation.

2. Find the Complementary Function ($y_c$):
For the homogeneous part, we assume a solution of the form $y=x^m$.
Substituting $y=x^m$, $\frac{dy}{dx}=m{x}^{m-1}$, and $\frac{d^{2}y}{d{x}^2}=m(m-1)x^{m-2}$ into the homogeneous equation yields:
$x^2\left[m\right(m-1\left)x^{m-2}\right]-2x\left[m{x}^{m-1}\right]+2x^m=0$
Factoring out $x^m$ (since $x^m\ne 0$), we obtain the auxiliary equation:
$m(m-1)-2m+2=0$
$m^2-3m+2=0$
Factoring the quadratic equation:
$(m-1)(m-2)=0$
Thus, the roots are $m=1$ and $m=2$.
The complementary function is:
$y_c=C_1{x}^2+C_{2}x$

3. Find the Particular Integral ($y_p$):
Since the right-hand side of the differential equation is a constant ($4$), we assume a particular solution of the form:
$y_p=A$
where $A$ is a constant. Taking derivatives:
$\frac{d{y}_p}{dx}=0$
$\frac{d^2{y}_p}{d{x}^2}=0$
Substitute these values back into the original differential equation:
$x^2\left(0\right)-2x\left(0\right)+2A=4$
$2A=4\Rightarrow A=2$
Therefore, the particular integral is:
$y_p=2$

4. Combine to get the Complete Solution:
The general solution is the sum of the complementary function and the particular integral:
$y=y_c+y_p$
$y=C_1{x}^2+C_{2}x+2$