solve the equation.\n\frac{dy}{dx}=\frac{(xy)^2}{8}+y^2\nchoose 1 answer:\n(a) y = -\frac{x^2 + 16}{16 +…

Explanation:

Step1: Rewrite the differential equation

We have $\frac{dy}{dx}=\frac{x^{2}y^{2}}{8}+y^{2}=y^{2}(\frac{x^{2}}{8} + 1)$. This is a first - order separable differential equation of the form $\frac{dy}{y^{2}}=(\frac{x^{2}}{8}+1)dx$.

Step2: Integrate both sides

Integrate $\int\frac{dy}{y^{2}}=\int(\frac{x^{2}}{8}+1)dx$. The left - hand side integral is $\int y^{-2}dy=-\frac{1}{y}+C_1$. The right - hand side integral: $\int(\frac{x^{2}}{8}+1)dx=\frac{1}{8}\int x^{2}dx+\int dx=\frac{1}{8}\times\frac{x^{3}}{3}+x + C_2=\frac{x^{3}}{24}+x + C_2$.

Step3: Solve for y

We have $-\frac{1}{y}=\frac{x^{3}}{24}+x + C$. Then $y =-\frac{24}{x^{3}+24x + C}$.

Answer:

D. $y =-\frac{24}{x^{3}+24x + C}$