For "Y" and what is the general solution]

y[2x³D³+12x²D²+14xDx+2]=[12+(4/x)]

Please Help me with details solution....this one is kinda hard for me

To find the general solution for y in the equation:

y[2x³D³ + 12x²D² + 14xDx + 2] = [12 + (4/x)]

Here's a step-by-step guide to solving this equation:

Step 1: Understand the notation
- The notation "D" represents a derivative. So, D³ means the third derivative, D² means the second derivative, and Dx means the derivative with respect to x.
- The notation y[f(x)] is just applying the function f(x) to the variable y.

Step 2: Expand the equation
- Write out the equation in expanded form using the product rule for differentiation:

2x³D³(y) + 12x²D²(y) + 14xDx(y) + 2y = 12 + (4/x)

Step 3: Simplify the equation
- If we differentiate y with respect to x, we get Dy(y) or y'.
- We can simplify the equation by substituting Dy for Dx(y):

2x³D³(y) + 12x²D²(y) + 14xDy(y) + 2y = 12 + (4/x)

Step 4: Look for a homogeneous solution
- A homogeneous solution is a solution to the equation when the right-hand side is zero (i.e., 12 + (4/x) = 0).
- In this case, there is no homogeneous solution because the right-hand side is not zero.

Step 5: Solve for the particular solution
- To find the particular solution, we need to solve for y in terms of x.
- Rearrange the equation to isolate y:

2x³D³(y) + 12x²D²(y) + 14xDy(y) - 10y = 12 + (4/x)

Step 6: Use the variation of parameters method
- The variation of parameters method can help us find the particular solution.
- Assume that the particular solution can be expressed as y = u(x)v(x), where u(x) and v(x) are functions of x.
- Substitute y = u(x)v(x) into the rearranged equation from Step 5.

2x³D³(uv) + 12x²D²(uv) + 14xD(uv) - 10(uv) = 12 + (4/x)

Step 7: Simplify the equation
- Expand the derivatives using the product rule, and simplify the equation:

2x³[u'''v + 3x²u''v + 3x³u'v' + x³uv'''] + 12x²[u''v + 2xu'v + x²uv''] + 14x[u'v + xu'v' + xu''v'] - 10(uv) = 12 + (4/x)

Step 8: Group the terms
- Group the terms of the equation according to the powers of x, and simplify further:

2x³[u'''v] + 6x⁴[u''v' + uv''] + 12x²[3u''v + 2xuv' + x²uv''] + 14x[u'v + xu''v' + xu'v'] - 10(uv) = 12 + (4/x)

Step 9: Solve for u(x) and v(x)
- Equate the coefficients of each power of x to zero and solve for the functions u(x) and v(x):

Coefficient of x³: 2[u'''v] = 0
Coefficient of x⁴: 6[u''v' + uv''] = 0
Coefficient of x²: 12[3u''v + 2xuv' + x²uv''] = 0
Coefficient of x: 14[u'v + xu''v' + xu'v'] = 0
Coefficient of 1: -10(uv) = 12

Step 10: Solve for u(x)
- Solve the differential equations obtained in the previous step for u(x) by integrating:

Coefficient of x³: 2[u'''] = 0 (since u'''v = 0)
=> u''' = 0
=> u'' = C₁ (where C₁ is a constant)
=> u' = C₁x + C₂ (where C₂ is another constant)
=> u = (C₁/2)x² + C₂x + C₃ (where C₃ is another constant)

Step 11: Solve for v(x)
- Substitute the expressions for u(x) obtained in Step 10 into the equations for v(x) from Step 9 and solve for v(x):

Coefficient of x⁴: 6[(C₁/2)x²v' + (C₁/2)xv''] = 0
=> xv'' + v' = 0 (dividing by (C₁/6)x²)
This is a first-order linear ordinary differential equation.

Step 12: Solve the equation for v(x)
- We can solve the equation for v(x) using standard methods for solving first-order linear ODEs.
- The solution for v(x) will depend on the specifics of the equation and any given initial or boundary conditions.

Step 13: Write the general solution
- Once we have the solutions for u(x) and v(x), we can write the general solution for y as:

y = (C₁/2)x²v(x) + C₂xv(x) + C₃v(x)

Note: The specific solution for v(x) will determine the form of the general solution.