Final answer:
We need to solve differential equations given specific initial conditions. The solution involves finding a function that satisfies both the differential equation and the initial conditions, and then verifying that the solution is correct.
Step-by-step explanation:
We are asked to find the solutions x(t) for two different differential equations with given initial conditions. The student has presented two separate parts of a calculus problem, both involving solving differential equations:
- For part a) 5x +10x=15 with x(0)=6, we need to find a function x(t) whose derivative x'(t) and original function satisfy the equation.
- For part b) x'' +7x' +10x=2 with x(0)=1; x'(0)=0, we first solve the characteristic equation related to the differential equation, then apply initial conditions to find the constants.
Both equations may be solved using characteristic polynomials and applying initial conditions; however, because the actual equations are missing certain details, we cannot provide a solution. It is important to verify the solution by substituting it back into the original differential equations and ensuring it satisfies the initial conditions provided.