Question

The ivp

sin(t)d²x/dt²+ cos(t)dx/dt +sin(t)x=tan(t)
x(1.25)=10
dx/dt∣∣₁.₂₅=10
has a unique solution defined on the interval ________

Answer 1

The question refers to a second-order non-homogeneous differential equation with given initial conditions. A unique solution exists on an interval around the initial point if the equation's coefficients are continuous there, but the exact interval will depend on the continuity of the trigonometric functions involved.

Step-by-step explanation:

The student's question pertains to solving a second-order non-homogeneous differential equation with initial value conditions. Unfortunately, the question contains several typos and unrelated segments, making it difficult to provide an exact answer for the interval on which the unique solution is defined. However, under the standard existence and uniqueness theorem for differential equations, if the coefficients of the differential equation and the non-homogeneous term are continuous on an interval, and the initial conditions are given at a point within this interval, then there exists a unique solution to the problem on some interval containing the initial point.

In this case, assuming that the functions ivp sin(t), cos(t), and tan(t) are continuous around t=1.25, and the initial conditions x(1.25)=10 and dx/dt at t=1.25=10 are provided, there would exist a unique solution to the differential equation on some interval that includes t=1.25. The exact interval would depend on the specifics of the equation and the domain of the trigonometric coefficients.