Questions: Find a general solution to the given equation for t<0.
y''(t) - (3/t) y'(t) + (29/t^2) y(t) = 0
Transcript text: Find a general solution to the given equation for $\mathrm{t}<0$.
\[
y^{\prime \prime}(t)-\frac{3}{t} y^{\prime}(t)+\frac{29}{t^{2}} y(t)=0
\]
Solution
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)=tm 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.
Step 1: Characteristic Equation
We start with the characteristic equation derived from the assumption y(t)=tm:
m(m−1)−3m+29=0
This simplifies to:
m2−4m+29=0
Step 2: Solve for m
Using the quadratic formula m=2a−b±b2−4ac, we find the roots:
m=2⋅14±(−4)2−4⋅1⋅29=24±16−116=24±−100
This gives us:
m=2±5i
Step 3: General Solution
Since the roots are complex, the general solution for the differential equation is given by:
y(t)=t2(C1cos(5ln∣t∣)+C2sin(5ln∣t∣))
where C1 and C2 are arbitrary constants.
Final Answer
The general solution is:
y(t)=t2(C1cos(5ln∣t∣)+C2sin(5ln∣t∣))