Question

Solve the separable differential equation dx/dt=x²+1/36 and find the particular solution satisfying the initial condition x(0)=2. Find x(t)

Answer 1

To solve the given separable differential equation with an initial condition, we separate the variables, integrate both sides, and then apply the initial condition to find the particular solution for x(t).

Step-by-step explanation:

To solve the separable differential equation dx/dt = x² + 1/36 and find the particular solution that satisfies the initial condition x(0) = 2, we start by separating the variables x and t. Then we integrate both sides of the equation with respect to their respective variables.

Step-by-step Solution:

1. Separate variables: dx / (x² + 1/36) = dt
2. Integrate both sides: ∫ dx / (x² + 1/36) = ∫ dt.
3. To find the particular solution, apply the initial condition x(0) = 2.

The step involving the integration and application of the initial condition is complex. However, once the integration is done, a function for x(t) is obtained that will satisfy the initial condition. This equation can be complicated and may need implicit function solving techniques depending on the resulting form after integration.