Final answer:
To solve the boundary value problem p'' + 6p' + 10p = 0 with the conditions p(0) = 0 and p(π/2) = 4, we find the roots of the characteristic equation, form the general solution based on those roots, and use the boundary conditions to determine the constants for the specific solution.
Step-by-step explanation:
The boundary value problem in question requires a solution to the second-order differential equation p'' + 6p' + 10p = 0 with the boundary conditions p(0) = 0 and p(π/2) = 4.
To solve this, we must find a general solution to the homogeneous differential equation and then apply the given boundary conditions to determine the specific solution that satisfies them.
The characteristic equation derived from the differential equation is r^2 + 6r + 10 = 0, which we can solve using the quadratic formula to find the roots of the characteristic equation.
These roots will be complex numbers due to the discriminant being negative. After finding the roots, the general solution will have the form C1e^(αx)cos(βx) + C2e^(αx)sin(βx), where α and β are the real and imaginary parts of the roots, respectively, and C1 and C2 are constants that we solve for using the boundary conditions.
Upon finding the general solution, we can plug in the boundary conditions one by one to find a system of equations. Setting p(0) = 0 gives us one equation in terms of the constants and setting p(π/2) = 4 gives us a second equation. Solving this system will give us the values of C1 and C2, thus providing the specific solution that satisfies the boundary value problem.