[Math] Difficult Homogeneous Differential Equation

integrationordinary differential equations

Solve the differential equation:

$$\frac{dy}{dx}=\frac{\sqrt{x^2+3xy+4y^2}}{x+2y}$$

I tried to solve it by putting $t=x+2y$ but that lead to a very complicated integral. The hint given is that equation is reducible to homogeneous form.

Best Answer

I suppose a typo and that the problem is $$\frac{dy}{dx}=\frac{\sqrt{x^2+\color{red}{4}xy+4y^2}}{x+2y}$$ Doing the same steps as in NAC's answer, we get $$\frac{dx}x=\frac {du}{1-u}$$ taht is to say

$$\log(x)+C=-\log(1-u)$$

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