We are tasked with finding a second solution $y_2(t)$ to the second-order differential equation:

$-5t^2y'' + 4ty' + 2y = 0$

Given that one solution $y_1(t) = t^2$, we can use the reduction of order method to find another linearly independent solution $y_2(t)$.

Steps for the Reduction of Order Method

1. Assume the form of the second solution $y_2(t)$: $y_2(t) = v(t) y_1(t)$ where $v(t)$ is a function to be determined, and $y_1(t) = t^2$ is the known solution.

2. Compute derivatives of $y_2(t)$: $y_2'(t) = v'(t) t^2 + v(t) \cdot 2t$ $y_2''(t) = v''(t) t^2 + 2v'(t)t + v'(t) \cdot 2t + v(t) \cdot 2$ Simplifying the second derivative: $y_2''(t) = v''(t) t^2 + 4v'(t)t + 2v(t)$

3. Substitute $y_2(t)$, $y_2'(t)$, and $y_2''(t)$ into the original ODE: $-5t^2 \left( v''(t) t^2 + 4v'(t)t + 2v(t) \right) + 4t \left( v'(t) t^2 + 2v(t)t \right) + 2 \left( v(t) t^2 \right) = 0$ Expanding the terms: $-5t^4v''(t) - 20t^3v'(t) - 10t^2v(t) + 4t^3v'(t) + 8t^2v(t) + 2t^2v(t) = 0$ Simplifying: $-5t^4v''(t) - 16t^3v'(t) = 0$ This simplifies to: $v''(t) + \frac{16}{5t}v'(t) = 0$

4. Solve for $v(t)$: This is a first-order differential equation in $v'(t)$. Let $u = v'(t)$, so the equation becomes: $u'(t) + \frac{16}{5t}u(t) = 0$ This is a separable equation. Solving it: $\frac{du}{u} = -\frac{16}{5t} dt$ Integrating both sides: $\ln |u| = -\frac{16}{5} \ln |t| + C_1$ Exponentiating: $u(t) = C_2 t^{-\frac{16}{5}}$ Thus, $v'(t) = C_2 t^{-\frac{16}{5}}$ Integrating again to find $v(t)$: $v(t) = \frac{C_2}{1 - \frac{16}{5}} t^{1 - \frac{16}{5}} + C_3 = \frac{C_2}{-\frac{11}{5}} t^{-\frac{11}{5}} + C_3$ Simplifying: $v(t) = -\frac{5}{11} C_2 t^{-\frac{11}{5}} + C_3$

5. Final form of $y_2(t)$: Recall that $y_2(t) = v(t) t^2$, so: $y_2(t) = \left( -\frac{5}{11} C_2 t^{-\frac{11}{5}} + C_3 \right) t^2 = -\frac{5}{11} C_2 t^{\frac{-11+10}{5}} + C_3 t^2$ $y_2(t) = -\frac{5}{11} C_2 t^{-\frac{1}{5}} + C_3 t^2$

6. Apply initial conditions $y_2(1) = -\frac{5}{11}$ and $y_2'(1) = \frac{1}{11}$ to determine constants $C_2$ and $C_3$.

Would you like to proceed with solving for these constants, or is any clarification needed on the process so far?

Related Questions:

1. How do we handle non-homogeneous ODEs using reduction of order?
2. Why does the reduction of order method work only when one solution is known?
3. How do initial conditions affect the form of the second solution?
4. What if the first solution wasn't $t^2$, how would the process change?
5. What are the limitations of reduction of order for higher-order ODEs?

Tip:

Reduction of order is powerful because it transforms a second-order ODE into a first-order ODE, which is generally easier to solve.