118k views
4 votes
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

User Dubmojo
by
8.8k points

1 Answer

2 votes

Final answer:

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.

User Wilson Delgado
by
8.0k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.