The equation given is:

$t Y'' + (1 - 2t) Y' - 2Y = 0,$

with initial conditions:

$Y(0) = 1, \quad Y'(0) = 2.$

This is a second-order linear differential equation with variable coefficients. To solve this, the following methods can be applied:

1. Determine if the equation is a Cauchy-Euler type:
   Rewrite the equation and check if substitutions can simplify the coefficients.

2. Power Series Solution:
   As the coefficients are functions of $t$, a power series expansion might be necessary, especially near $t = 0$.

3. Numerical Methods:
   Depending on whether a closed-form solution is obtainable, numerical methods may provide approximations for the initial conditions.

I will analyze this equation to find the general or exact solution. Let me compute that for you.from sympy import symbols, Function, Eq, dsolve, Derivative, solve

Define variables and function

t = symbols('t') Y = Function('Y')(t)

Define the differential equation

diff_eq = Eq(t * Y.diff(t, 2) + (1 - 2 * t) * Y.diff(t) - 2 * Y, 0)

Define initial conditions

initial_conditions = {Y.subs(t, 0): 1, Y.diff(t).subs(t, 0): 2}

Solve the differential equation

solution = dsolve(diff_eq, Y, ics=initial_conditions) solutionEq(Y(t), C1*(4t**5/15 + 2t4/3 + 4*t3/3 + 2t**2 + 2t + 1) + O(t6))Eq(Y(t), C1*(4*t5/15 + 2t**4/3 + 4t3/3 + 2*t2 + 2*t + 1) + O(t**6))successThe solution to the given differential equation using a power series approximation is:

$Y(t) = C_1 \left(\frac{4t^5}{15} + \frac{2t^4}{3} + \frac{4t^3}{3} + 2t^2 + 2t + 1\right) + O(t^6),$

where $C_1$ is a constant from the power series expansion, and $O(t^6)$ represents higher-order terms.

Would you like further details on the solution process or refinement of this series? Here are a few related questions you might explore:

1. How are power series solutions derived for differential equations?
2. Why do we use a power series instead of a general closed-form solution for variable coefficient equations?
3. Can this equation be solved numerically for any specific range of $t$?
4. What is the significance of $O(t^6)$, and how do we interpret it in practical applications?
5. Are there special cases where exact solutions for such equations exist without series expansion?

Tip: When solving differential equations with variable coefficients, always check for symmetry or simplifications to avoid lengthy computations.