The solution of the differential equation $\frac{dy}{dx}=9\frac{x}{y}$ represents

Solving the Differential Equation dy/dx = 9x/y

The question asks us to identify the geometric shape represented by the solution to the differential equation:
$\frac{dy}{dx}=9\frac{x}{y}$

Step-by-Step Solution

This is a classic example of a separable differential equation. We can solve it by separating the variables:

1. Separate Variables: Rearrange the equation so that all terms involving $y$ are on one side (with $dy$) and all terms involving $x$ are on the other side (with $dx$). Multiply both sides by $y$ and $dx$: $ydy=9xdx$
2. Integrate Both Sides: Now, integrate both sides of the equation with respect to their respective variables. $\int ydy=\int 9xdx$
3. Perform Integration: Use the power rule for integration ($\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$). $\frac{y^2}{2}=9\left(\frac{x^2}{2}\right)+C_1$ Where $C_1$ is the constant of integration.
4. Simplify the Equation: To make the equation clearer, multiply the entire equation by 2. $y^2=9x^2+2C_1$ Let $C=2C_1$. Since $C_1$ is an arbitrary constant, $C$ is also an arbitrary constant. $y^2=9x^2+C$
5. Analyze the Resulting Equation: Rearrange the equation to identify the geometric shape. $y^2-9x^2=C$

Identifying the Geometric Shape

The equation $y^2-9x^2=C$ represents a conic section.

- If $C=0$, the equation becomes $y^2-9x^2=0$, which factors as $(y-3x)(y+3x)=0$. This represents two intersecting straight lines, $y=3x$ and $y=-3x$. This is considered a degenerate hyperbola.
- If $C\ne 0$, we can rewrite the equation. Assuming $C>0$: $\frac{y^2}{C}-\frac{9x^2}{C}=1$ $\frac{y^2}{(\sqrt{C}{)}^2}-\frac{x^2}{(\sqrt{C/9}{)}^2}=1$ This is the standard form of a hyperbola with the transverse axis along the y-axis.
- If $C<0$, let $C=-k$ where $k>0$. The equation becomes $y^2-9x^2=-k$, or $9x^2-y^2=k$. Dividing by $k$: $\frac{9x^2}{k}-\frac{y^2}{k}=1$ $\frac{x^2}{(\sqrt{k/9}{)}^2}-\frac{y^2}{(\sqrt{k}{)}^2}=1$ This is the standard form of a hyperbola with the transverse axis along the x-axis.

In all non-degenerate cases ($C\ne 0$), the equation $y^2-9x^2=C$ defines a hyperbola. Therefore, the solution of the given differential equation represents a hyperbola.

Conclusion

By separating variables and integrating, we arrive at the equation $y^2-9x^2=C$. This equation describes a hyperbola (or a pair of intersecting lines if $C=0$, which is a degenerate case of a hyperbola).