I am trying to find the general solution of
$\displaystyle y(dy/dx) + x = sqrt(x^2 + y^2)
$
I'm having issues figuring out what n should be, as I can't get the y's together to figure out the powers. How would I do this.
thanks
I am trying to find the general solution of
$\displaystyle y(dy/dx) + x = sqrt(x^2 + y^2)
$
I'm having issues figuring out what n should be, as I can't get the y's together to figure out the powers. How would I do this.
thanks
$\displaystyle y\frac{dy}{dx} + x = \sqrt{x^2 + y^2}$
$\displaystyle \Rightarrow \frac{dy}{dx} + \frac{x}{y} = \frac{\sqrt{x^2 + y^2}}{y}$
$\displaystyle \Rightarrow \frac{dy}{dx} + \frac{x}{y} = \sqrt{\frac{x^2 + y^2}{y^2}}$
$\displaystyle \Rightarrow \frac{dy}{dx} + \frac{x}{y} = \sqrt{\left( \frac{x}{y} \right)^2 + 1}$
The standard technique is to now make the substitution $\displaystyle y = vx \Rightarrow \frac{dy}{dx} = v + \frac{dv}{dx}$. After some simplifying you get
$\displaystyle \frac{dv}{dx} + 1 + v^2 = \sqrt{1 + v^2}$.
This DE is seperable, leading to $\displaystyle x = \frac{\sqrt{v^2 + 1} + 1}{v} + C$. Now substitute back $\displaystyle y = vx \Rightarrow v = \frac{y}{x}$ .....