Question

Solve the differential equation by variation of parameters.
y'' + y = 1/sin x

Answer 1

To solve the differential equation y'' + y = 1/sin(x) by variation of parameters, you need to find the complementary and particular solutions and then combine them to obtain the general solution.

Step-by-step explanation:

To solve the given differential equation, y'' + y = 1/sin(x), by variation of parameters, we need to follow these steps:

1. Find the complementary solution by solving the associated homogeneous equation, y'' + y = 0. The complementary solution is yc = A*cos(x) + B*sin(x).
2. Find the particular solution by assuming yp = u1(x)*cos(x) + u2(x)*sin(x), where u1(x) and u2(x) are unknown functions.
3. Differentiate yp twice and substitute it into the given differential equation to obtain a system of equations for u1' and u2'.
4. Solve the system of equations obtained in the previous step to find u1' and u2'.
5. Integrate u1' and u2' to get u1(x) and u2(x).
6. The general solution to the original differential equation is y(x) = yc + yp = A*cos(x) + B*sin(x) + u1(x)*cos(x) + u2(x)*sin(x).