Find the general solution of the differential equation whose…

Find the general solution of the differential equation whose solution is given by $y = C e^{\sqrt{x^{2}-1}}$.

Step 1: Eliminate the constant

Differentiating $y=Ce^{\sqrt{x^2-1}}$ using the chain rule, with $\dfrac{d}{dx}\sqrt{x^2-1}=\dfrac{x}{\sqrt{x^2-1}}$ for $|x|>1$, gives

\[\frac{dy}{dx}=Ce^{\sqrt{x^2-1}}\cdot\frac{x}{\sqrt{x^2-1}}=\frac{x}{\sqrt{x^2-1}}\,y.\]

\[\boxed{\,y'=\dfrac{x}{\sqrt{x^2-1}}\,y,\qquad |x|>1.}\]

Step 2: Solve the ODE by separation of variables

Separating variables, $\dfrac{dy}{y}=\dfrac{x}{\sqrt{x^2-1}}\,dx$. With $u=x^2-1$, $du=2x\,dx$, the right side integrates to $\sqrt{x^2-1}$, so

\[\ln|y|=\sqrt{x^2-1}+C_1 \quad\Longrightarrow\quad y=C_0\,e^{\sqrt{x^2-1}},\qquad |x|>1,\]

where $C_0=\pm e^{C_1}$ (also allowing $C_0=0$), reproducing the original family.

Step 3: Domain and verification

The expression $\sqrt{x^2-1}$ is real only for $|x|>1$, which is the natural domain of every solution curve. Differentiating $y=C_0e^{\sqrt{x^2-1}}$ again reproduces $y'=\dfrac{x}{\sqrt{x^2-1}}y$, confirming the solution.