Question

Determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. Do not attempt to find the solution. y′′+(cost)y′+3(ln∣t∣)y=0, y(2)=3, y ′(2)=1

Answer 1

The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is (-∞, ∞).

Step-by-step explanation:

To determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution, we need to examine the coefficients of the differential equation. In this case, the coefficient of the second derivative term (y'') is 1, the coefficient of the first derivative term (y') is cos(t), and the coefficient of the y term is 3(ln|t|).

Since these coefficients are continuous and bounded on the interval (-∞, ∞) (since cos(t) and ln|t| are continuous and bounded for all t), the given initial value problem is certain to have a unique twice-differentiable solution on this interval.