Question

Determine the largest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. Do not attempt to find the solution.

sin(t)d²x/dt²+cos(t)dx/dt+sin(t)x=tan(t) ,x(1)=20,x′(1)=5

Answer 1

The largest interval for the initial value problem with a unique twice-differentiable solution is from (0, π/2), taking into account the continuity and non-zero nature of the trigonometric functions involved.

Step-by-step explanation:

The initial value problem consists of a second-order differential equation with initial conditions. To ensure a unique twice-differentiable solution, we look for regularity and smoothness in the coefficients of the differential equation as well as in the non-homogeneous part (on the right side of the equation).

Given the equation sin(t)d²x/dt² + cos(t)dx/dt + sin(t)x = tan(t), we must consider the continuity of sin(t), cos(t), and tan(t). Since tan(t) is undefined when cos(t) is zero, we look for intervals where cos(t) ≠ 0. The largest such interval containing t=1 is (0, π/2). Therefore, in this interval, the problem is guaranteed to have a unique solution by the existence and uniqueness theorem for differential equations.