find the solution of the differential equation that satisfies the given initial condition.\n\\( \\frac{d…

Explanation:

Step1: Separate variables

Separate the variables in the differential equation (\frac{dy}{dx}=\frac{x}{y}). Multiply both sides by (y) and (dx) to get (y;dy = x;dx).

Step2: Integrate both sides

Integrate (\int y;dy=\int x;dx). Using the power - rule (\int t^n;dt=\frac{t^{n + 1}}{n+1}+C) ((n\neq - 1)), we have (\frac{y^{2}}{2}=\frac{x^{2}}{2}+C).

Step3: Use the initial condition

Given (y(0)=-9), substitute (x = 0) and (y=-9) into (\frac{y^{2}}{2}=\frac{x^{2}}{2}+C). (\frac{(-9)^{2}}{2}=\frac{0^{2}}{2}+C), so (C=\frac{81}{2}).

Step4: Solve for (y)

Substitute (C=\frac{81}{2}) back into (\frac{y^{2}}{2}=\frac{x^{2}}{2}+C). (\frac{y^{2}}{2}=\frac{x^{2}}{2}+\frac{81}{2}), then (y^{2}=x^{2}+81), and (y=\pm\sqrt{x^{2}+81}). Since (y(0)=-9), we take the negative square - root.

Answer:

(y =-\sqrt{x^{2}+81})