Question

Solve the initial value problem.
dy/dx = (1 + y²) tan x; y(0) = √3

Answer 1

To solve the initial value problem dy/dx = (1 + y²) tan x; y(0) = √3, we can separate the variables and integrate both sides. Rearranging the equation, we have 1/(1 + y²) dy = tan x dx. Integrating both sides gives us arctan(y) = ln|sec x| + C. Now we can solve for y by taking the tangent of both sides, resulting in y = tan(arctan(y)) = tan(ln|sec x| + C). Finally, we substitute the initial condition y(0) = √3 to find the value of C and obtain the solution of the initial value problem.

Step-by-step explanation:

To solve the initial value problem dy/dx = (1 + y²) tan x with the initial condition y(0) = √3, we can separate the variables and integrate both sides. Rearranging the equation, we have 1/(1 + y²) dy = tan x dx. Integrating both sides gives us arctan(y) = ln|sec x| + C. Now we can solve for y by taking the tangent of both sides, resulting in y = tan(arctan(y)) = tan(ln|sec x| + C). Finally, we substitute the initial condition y(0) = √3 to find the value of C and obtain the solution of the initial value problem.