Incompatible solutions of differential equation

Question

The solution of the differential equation $(\frac{dy}{dx})^2=(1-y^2)(1-k^2+k^2 y^2)$ for the initial condition $y(0)=1$ is $y(x)=cn(x, k^2)$ where $cn$ is Jacobi elliptic function. I found the general solution of this equation is $y(x)=\sqrt{1-\frac{1}{k^2 sn^2(x+c1, k^2)}}$ where c1 is the integration constant. But when I use the initial condition $y(0)=1$ in it, I can't get a valid value of integration constant c1.

Answer 1

$$(\frac{dy}{dx})^2=(1-y^2)(1-k^2+k^2 y^2)$$ The general solution is $y(x)=cn(x+c_1\:,\:k^2)$

with the initial condition $y(0)=1\quad\to\quad c_1=0$ $$y(x)=cn(x,k^2)$$ The relationship between $cn$ and $sn$ is : $cn^2(x,k^2)=1-sn^2(x,k^2)$ , thus : $$y(x)=\sqrt{1-sn^2(x+c_1\:,\:k^2)}$$ $y(x)=\sqrt{1-\frac{1}{k^2 sn^2(x+c1, k^2)}}$ is false.

Finally, with the initial condition : $$y(x)=\sqrt{1-sn^2(x,k^2)}$$ $sn(0,k^2)=0 \quad\to\quad y(0)=1$ .