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