Solve the following differential equations:
(I). Cos x dy/dx + ysin x= sec^2 x
(II). Y"-10y' + 41y=0
To solve the given differential equations, we will follow these steps:
(I). Cos x dy/dx + ysin x = sec^2 x
Step 1: Rewrite the given equation in standard form
Multiply through by dx and divide by cos x to obtain:
dy = (sec^2 x)/cos x dx
Simplify:
dy = sec x dx
Step 2: Integrate both sides
Integrating both sides:
∫ dy = ∫ sec x dx
Integrating the left side:
y = ∫ sec x dx
Integrating the right side requires applying the formula for the integral of sec x:
y = ln |sec x + tan x| + C
where C is the constant of integration.
(II). y'' - 10y' + 41y = 0
Step 1: Write the characteristic equation
We assume a solution of the form y = e^(rt), where r is a constant.
Substitute this into the differential equation:
r^2 e^(rt) - 10r e^(rt) + 41 e^(rt) = 0
Step 2: Solve the characteristic equation
Divide through by e^(rt):
r^2 - 10r + 41 = 0
This is a quadratic equation, which can be solved using the quadratic formula:
r = (10 ± √(10^2 - 4(1)(41)) / (2)
Simplifying further:
r = (10 ± √(-164)) / 2
r = (10 ± √(164)i) / 2
r = 5 ± √(41)i
So the solutions are complex conjugates.
Step 3: Find the general solution
Since the roots are complex, the general solution will have the form:
y = e^(5t) (C1 cos √(41)t + C2 sin √(41)t)
where C1 and C2 are arbitrary constants.
These are the solutions to the given differential equations.