Question:

Step 1: To solve the given differential equation $(x^2+y^2)\,dx-5xy\,dy=0$, we first need to identify it as a homogeneous differential equation, which can be solved using the substitution $y = vx$, where $v$ is a function of $x$. This step involves recognizing the form of the equation and applying the appropriate method for solving it. Step 2: The given differential equation can be rearranged as $\frac{dy}{dx} = \frac{x^2+y^2}{5xy}$. By substituting $y = vx$, we can find $\frac{dy}{dx} = v + x\frac{dv}{dx}$. Substituting these into the rearranged equation gives us $v + x\frac{dv}{dx} = \frac{x^2+(vx)^2}{5x(vx)} = \frac{1+v^2}{5v}$. This simplification allows us to separate the variables. Step 3: Separating the variables in the equation $v + x\frac{dv}{dx} = \frac{1+v^2}{5v}$ yields $x\frac{dv}{dx} = \frac{1+v^2}{5v} - v = \frac{1+v^2-5v^2}{5v} = \frac{1-4v^2}{5v}$. This leads to the equation $x\frac{dv}{dx} = \frac{1-4v^2}{5v}$, which can be further simplified to $\frac{5v}{1-4v^2}dv = \frac{dx}{x}$. Step 4: Integrating both sides of the equation $\frac{5v}{1-4v^2}dv = \frac{dx}{x}$ gives us $\int\frac{5v}{1-4v^2}dv = \int\frac{dx}{x}$. Using the substitution $u = 1 - 4v^2$, we find $du = -8v\,dv$, or $v\,dv = -\frac{1}{8}du$. Thus, the integral becomes $-\frac{5}{8}\int\frac{du}{u} = \int\frac{dx}{x}$, which simplifies to $-\frac{5}{8}\ln|u| = \ln|x| + C$, where $C$ is the constant of integration. Step 5: Substituting back $u = 1 - 4v^2$ and $v = \frac{y}{x}$ into the equation $-\frac{5}{8}\ln|u| = \ln|x| + C$ yields $-\frac{5}{8}\ln|1-4\frac{y^2}{x^2}| = \ln|x| + C$. Simplifying, we get $-\frac{5}{8}\ln|1-4\frac{y^2}{x^2}| - \ln|x| = C$, which can be rewritten as $-\frac{5}{8}\ln|1-4\frac{y^2}{x^2}| - \ln|x| = C$. Using properties of logarithms, we can combine the terms to obtain $-\frac{5}{8}\ln|1-4\frac{y^2}{x^2}| - \ln|x| = \ln|x|^{-\frac{5}{8}} - \ln|x| = \ln\left(\frac{1}{|x|^{\frac{5}{8}}}\right) - \ln|x| = \ln\left(\frac{1}{|x|^{\frac{5}{8}+1}}\right) = \ln\left(\frac{1}{|x|^{\frac{13}{8}}}\right)$. Step 6: However, the correct approach to solve the given differential equation should directly lead to the solution without unnecessary complications. The equation can be simplified by recognizing it as a homogeneous equation and applying the appropriate substitution directly. The correct solution should directly address the given differential equation and apply the initial condition $y(1) = 0$ to find the specific solution. Given that, we should re-evaluate our approach to ensure it aligns with solving the differential equation $(x^2+y^2)\,dx-5xy\,dy=0$ with the condition $y(1)=0$, and directly obtain the solution from the provided options. Step 7: Revisiting the differential equation and considering the initial condition $y(1) = 0$, we aim to find a solution that satisfies both the differential equation and the given condition. The correct solution should be derived by solving the differential equation and applying the initial condition to determine the specific solution among the provided options. Step 8: The solution of the differential equation, considering the initial condition $y(1)=0$, is given by $|x^2-4y^2|^5=x^2$. This solution satisfies the differential equation $(x^2+y^2)\,dx-5xy\,dy=0$ and the initial condition. Therefore, the correct answer is the one that matches this solution. The final answer is: $|x^2-4y^2|^5=x^2$, which corresponds to Option 3.