Differential equations [arctan(x) to tan(x)]

[says]

Homework Statement

dx/dt = (1+x²)e^t ; x(0) = 1

1/1+x² dx/dt = e^t
∫ 1/1+x² (dx/dt) * dt = ∫ e^t dt
∫ 1/(1+x²) dx = ∫ e^t dt
arctan(x) = e^t + c
so x(t) = tan (e^t+c)

Homework Equations

The Attempt at a Solution

dx/dt = (1+x²)e^t ; x(0) = 1

1/1+x² dx/dt = e^t
∫ 1/1+x² (dx/dt) * dt = ∫ e^t dt
∫ 1/(1+x²) dx = ∫ e^t dt

arctan(x) = e^t + c
so x(t) = tan (e^t+c)

I'm a bit confused by the last step. How does arctan(x) = e^t + c become x(t) = tan (e^t+c). Cheers!

 

[Fredrik]

I'm a bit confused by the last step. How does arctan(x) = e^t + c become x(t) = tan (e^t+c). Cheers!

For any function f, if x is in the domain of f and x=y, then f(x)=f(y). Also, we have tan(arctan z)=z for all z in the domain of arctan.

By the way, you shouldn't write 1/(1+x²) as 1/1+x². The latter is equal to 1+x².