- #1

zenterix

- 555

- 76

- Homework Statement
- I am working on differential equations and in my calculations I ran into the sequence of expressions

$$\sqrt{1+c}=\sqrt{-|1+c|}=i\sqrt{|1+c|}=i\sqrt{-(1+c)}=i^2\sqrt{1+c}=-\sqrt{1+c}$$

where ##c<-1##.

- Relevant Equations
- Is the sequence of expression above correct? It does not seem so.

Now, it is very likely that I am making a super silly mistake here. But I can't see it.

For context, I have a 2x2 system of linear first order differential equations

$$\vec{x}'=A\vec{x}$$

where

$$A=\begin{bmatrix} 0 & 1\\c&-2\end{bmatrix}$$

and the characteristic polynomial is

$$\lambda^2+2\lambda-c=0$$

The two eigenvalues are ##-1\pm\sqrt{1+c}##.

Suppose ##c<-1##. Then ##1+c<0##.

So I was just in the middle of writing the eigenvalues given these values of ##c##.

$$\vec{x}'=A\vec{x}$$

where

$$A=\begin{bmatrix} 0 & 1\\c&-2\end{bmatrix}$$

and the characteristic polynomial is

$$\lambda^2+2\lambda-c=0$$

The two eigenvalues are ##-1\pm\sqrt{1+c}##.

Suppose ##c<-1##. Then ##1+c<0##.

So I was just in the middle of writing the eigenvalues given these values of ##c##.