Question

Solve the separable differential equation dx/dt=x²+(1/16) and find the particular solution satisfying the initial condition x(0)=9

Answer 1

To solve the separable differential equation dx/dt=x²+(1/16), we can separate the variables and integrate both sides. The particular solution satisfying the initial condition x(0)=9 is x(t) = 1/(4t+1/36).

Step-by-step explanation:

To solve the separable differential equation dx/dt=x²+(1/16), we can separate the variables and integrate both sides. Starting with dx/(x²+(1/16))=dt, we can integrate the left side using the integral of a fraction with a linear denominator. This yields 1/4(4x+1)/(x²+(1/16)) = t + C. To find the particular solution satisfying the initial condition x(0)=9, we substitute the values into the equation and solve for C. Plugging in x=9 and t=0 gives us C=1/36. Therefore, the particular solution is x(t) = 1/(4t+1/36).