Questions: Find a general solution to the given equation for t<0. y''(t) - (3/t) y'(t) + (29/t^2) y(t) = 0

Solution Steps

To solve the given differential equation, we recognize it as a Cauchy-Euler equation. The general approach involves assuming a solution of the form \( y(t) = t^m \) and substituting it into the differential equation to find the characteristic equation. Solving the characteristic equation will give us the values of \( m \), which we use to construct the general solution.

We start with the characteristic equation derived from the assumption \( y(t) = t^m \): \[ m(m - 1) - 3m + 29 = 0 \] This simplifies to: \[ m^2 - 4m + 29 = 0 \]

Using the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we find the roots: \[ m = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 29}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 116}}{2} = \frac{4 \pm \sqrt{-100}}{2} \] This gives us: \[ m = 2 \pm 5i \]

Since the roots are complex, the general solution for the differential equation is given by: \[ y(t) = t^2 \left( C_1 \cos(5 \ln |t|) + C_2 \sin(5 \ln |t|) \right) \] where \( C_1 \) and \( C_2 \) are arbitrary constants.

Final Answer