<s>[INST] Differential equation : $y' = (x+1)/(xy+x)$ So, I have the following differential equation to solve : $$y' = \frac{x+1}{xy+x}$$
After several steps, I get here :
$t^2 + 2t = 2x + 2ln(x) + c$
How do I isolate $t ?$
thank you!
By the way, $ t=f(x)=y$
 [/INST] Start by dividing through by $x$:
$$
y'=\frac{1+\frac{1}{x}}{y+1} \\
y'(y+1)=1+\frac{1}{x} \\
\int y'(y+1)dx=\int 1+\frac{1}{x}dx \\
\frac{y^2}{2}+2y=2x+2\ln(x)+c~\text{for constant $c$} \\
y^2+2y+1=2x+2\ln(x)+2c+1 \\
(y+1)^2=2x+2\ln(x)+2c+1 \\
y+1=\pm \sqrt{2x+2\ln(x)+2c+1} \\
y=\pm \sqrt{2x+2\ln(x)+2c+1}-1.~_{\square}
$$
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