I have an example of Euler's numeric method:
$\frac{dy}{dx}=2xy$
$y(x)=2 $
So $f(x,y)=2xy $ and the step: $x_{i+1} = x_i +h $ where $h=0.1$
$y_{i+1} = y_i +h f(x_i,y_i) $
The first iteration is
$y_{1} = y_0 + h \cdot f(0,2)= 0 +0.1 \times (2 \times 0 \times 2)=2$
$x_{1} = x_0 + h = 0 +0.1 = 0.1 .$
The second iteration is
$ y_{2} = y_1 + h \times f(0.1,2)= 0 +0.1 \times (2 \times 0.1 \times 2)=2.4 $
$ x_{2} = x_1 + h = 0.1 +0.1 = 0.2 .$
etc...
My question is that I am facing a problem like:
$ \frac{dy}{dt} = \frac{t}{y} ,$ with $y(0)=1$ and $y(t)=\sqrt{1+t^{2}}$.
In the example, besides the obvious switch $(x,t)$, The $y(t)=\sqrt{1+t^{2}}$ is not given so to solve the problem I do not know hw to apply the Euler method Could you help in this one?