14. x(1+y^2)^{\frac{1}{2}}dx=y(1+x^2)^{\frac{1}{2}}dy solve separable method

14. x(1+y^2)^{\frac{1}{2}}dx=y(1+x^2)^{\frac{1}{2}}dy solve separable method.

To solve this differential equation by separable method, we need to separate the variables x and y on opposite sides of the equation and integrate both sides.

Starting with the given differential equation:

(1) $\begin{equation*}x(1+y^2)^{\frac{1}{2}}dx=y(1+x^2)^{\frac{1}{2}}dy\end{equation*}$

We can divide both sides by $(1+y^2)^{\frac{1}{2}}(1+x^2)^{\frac{1}{2}}$ to obtain:

(2) $\begin{equation*}\frac{x}{(1+x^2)^{\frac{1}{2}}}dx=\frac{y}{(1+y^2)^{\frac{1}{2}}}dy\end{equation*}$

Now we can integrate both sides with respect to their respective variables:

(3) $\begin{equation*}\int \frac{x}{(1+x^2)^{\frac{1}{2}}}dx=\int \frac{y}{(1+y^2)^{\frac{1}{2}}}dy\end{equation*}$

For the left-hand side integral, we can use the substitution $u = 1 + x^2$ , $du = 2x dx$ to obtain:

(4) $\begin{equation*}\int \frac{x}{(1+x^2)^{\frac{1}{2}}}dx = \frac{1}{2} \int \frac{du}{\sqrt{u}} = \sqrt{u} + C_1 = \sqrt{1+x^2} + C_1\end{equation*}$

where $C_1$ is the constant of integration.

For the right-hand side integral, we can use the substitution $v = 1 + y^2$ , $dv = 2y dy$ to obtain:

(5) $\begin{equation*}\int \frac{y}{(1+y^2)^{\frac{1}{2}}}dy = \frac{1}{2} \int \frac{dv}{\sqrt{v}} = \sqrt{v} + C_2 = \sqrt{1+y^2} + C_2\end{equation*}$