Solve the differential equation (dy)/(dx)=(3)/(xy^(2))

Separate Variables: We are given the differential equation (dy)/(dx)=(3)/(xy^(2)). To integrate this with respect to x, we need to separate the variables y and x.Rewrite Equation: Rewrite the equation to separate the variables: (dy)/(dx) = (3)/(xy^(2)) Multiply both sides by dx and divide by y^2 to get: dy/y^2 = (3)/(x) dxIntegrate Separately: Now, integrate both sides of the equation with respect to their respective variables: ∫(dy/y^2) = ∫((3)/(x)) dxApply Integrals: The integral of 1/y^2 with respect to y is -1/y, and the integral of 3/x with respect to x is 3ln|x|. So we have: -1/y = 3ln|x| + C, where C is the constant of integration.Final Solution: We have successfully separated the variables and integrated both sides. The solution to the differential equation is: -1/y = 3ln|x| + C

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Solve the differential equation (dy)/(dx)=(3)/(xy^(2))

Solve the differential equation $\frac{d y}{d x}=\frac{3}{x y^{2}}$

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Q. Solve the differential equation

\frac{d y}{d x}=\frac{3}{x y^{2}}

Separate Variables: We are given the differential equation $\frac{dy}{dx}=\frac{3}{xy^{2}}$ . To integrate this with respect to $x$ , we need to separate the variables $y$ and $x$ .
Rewrite Equation: Rewrite the equation to separate the variables: $\newline$ $\frac{dy}{dx} = \frac{3}{xy^{2}}$ $\newline$ Multiply both sides by dx and divide by $y^{2}$ to get: $\newline$ $\frac{dy}{y^{2}} = \frac{3}{x} dx$
Integrate Separately: Now, integrate both sides of the equation with respect to their respective variables: $\newline$ $\int(\frac{dy}{y^2}) = \int(\frac{3}{x}) dx$
Apply Integrals: The integral of $\frac{1}{y^2}$ with respect to $y$ is $-\frac{1}{y}$ , and the integral of $\frac{3}{x}$ with respect to $x$ is $3\ln|x|$ . So we have: $\newline$ $-\frac{1}{y} = 3\ln|x| + C$ , where $C$ is the constant of integration.
Final Solution: We have successfully separated the variables and integrated both sides. The solution to the differential equation is: $-\frac{1}{y} = 3\ln|x| + C$