Final answer:
To solve the initial value problem for the Cauchy Euler equation, substitute y(t) = t^r into the equation and solve for the roots of r. Then, substitute the initial conditions to find the constants in the solution. Finally, verify the solution by checking that it satisfies the initial conditions.
Step-by-step explanation:
To solve the given initial value problem for the Cauchy Euler equation, we can use the substitution method. Let's assume y(t) = t^r, where r is a constant to be determined. Substitute y(t) and its derivatives into the equation:
t^2(t^r) - 6t(t^r)' + 12(t^r) = 0
Simplify and solve for r:
r(r-1)t^r + 6r(t^(r-1)) + 12t^r = 0
This equation is a quadratic equation in terms of r. Solve for the roots of r and substitute them back into y(t) = t^r to get the two solutions:
y₁(t) = t^2
y₂(t) = t^3
Now, substitute the initial conditions y(1) = 2 and y'(1) = 3 into each solution to find the corresponding constants:
y₁(1) = 1² = 1, y₂(1) = 1³ = 1
From the second equation, y₁'(t) = 2t and y₂'(t) = 3t^2. Substitute t = 1 into each derivative:
y₁'(1) = 2(1) = 2, y₂'(1) = 3(1)² = 3
Therefore, the solution to the initial value problem is y(t) = t^2 when y(1) = 2 and y'(1) = 2.