# Thread: [SOLVED] Differential Equation General Solution Help

1. ## [SOLVED] Differential Equation General Solution Help

hey guys, i'm really struggling with this and need someone to point me in the right direction, any help at all would be greatly appreciated!

Find the general solution of Eq.2 and hence the general solution for y(x). Your answer should have 2 arbitrary constants of integration.

Here are the equation's i've got so far:

1. $\frac{dy}{dx}=\frac{{\rho}g}{T}\int^x_{0}\sqrt{1+\ frac{dy}{dt}^2}dt$

Differentiating both sides of Eq.1 to produce second order ODE for y(x)

$\frac{d^2y}{dx^2}=\frac{{\rho}g}{T}\sqrt{1+\frac{d y}{dx}^2}$

By letting $u=\frac{dy}{dx}$

2.
$\frac{du}{dx}=\frac{{\rho}g}{T}\sqrt{1+u^2}$

2. Note that your resulting equation is separable:
$\frac{du}{dx}=\frac{{\rho}g}{T}\sqrt{1+u^2}$
$\frac{du}{\sqrt{1+u^2}}=\frac{{\rho}g}{T}\,dx$
Integrate both sides and solve for u(x), remembering your constant of integration. Then, use u(x)=y'(x), and integrate to find y(x) (with your second constant of integration).

--Kevin C.