solve the equation. \n\frac{dy}{dx}=\frac{3}{xy^{2}}\nchoose 1 answer: \n(a) y = sqrt3{\frac{9x^{2}}{2}+c}\n(…

solve the equation. \n\frac{dy}{dx}=\frac{3}{xy^{2}}\nchoose 1 answer: \n(a) y = sqrt3{\frac{9x^{2}}{2}+c}\n(b) y=sqrt3{\frac{9x^{2}}{2}+c}\n(c) y=sqrt3{9ln|x|+c}\n(d) y=sqrt3{9ln|x + c|}

solve the equation. \n\frac{dy}{dx}=\frac{3}{xy^{2}}\nchoose 1 answer: \n(a) y = sqrt3{\frac{9x^{2}}{2}+c}\n(b) y=sqrt3{\frac{9x^{2}}{2}+c}\n(c) y=sqrt3{9ln|x|+c}\n(d) y=sqrt3{9ln|x + c|}

Answer

Explanation:

Step1: Separate variables

Separate $y$ - terms and $x$ - terms. We have $y^{2}dy=\frac{3}{x}dx$.

Step2: Integrate both sides

Integrate $\int y^{2}dy=\int\frac{3}{x}dx$. Using the power - rule $\int y^{n}dy=\frac{y^{n + 1}}{n+1}+C_1$ ($n\neq - 1$) and $\int\frac{1}{x}dx=\ln|x|+C_2$, we get $\frac{y^{3}}{3}=3\ln|x|+C$.

Step3: Solve for $y$

Multiply both sides by 3 to get $y^{3}=9\ln|x| + 3C$. Let $C'=3C$, then $y=\sqrt[3]{9\ln|x|+C'}$.

Answer:

C. $y = \sqrt[3]{9\ln|x|+C}$