Final answer to the problem
$-\frac{1}{3}\ln\left(\frac{y}{x}+1\right)+\frac{1}{3}\ln\left(\frac{y}{x}-2\right)=\ln\left(x\right)+C_0$
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Step-by-step Solution
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- Exact Differential Equation
- Linear Differential Equation
- Separable Differential Equation
- Homogeneous Differential Equation
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
- Integrate by substitution
- Integrate by parts
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1
Multiply both sides of the equation by $dx$
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$dy=\left(\frac{y^2}{x^2}-2\right)dx$
Learn how to solve problems step by step online. Solve the differential equation dy/dx=(y^2)/(x^2)-2. Multiply both sides of the equation by dx. Combine all terms into a single fraction with x^2 as common denominator. Divide both sides of the equation by dx. We can identify that the differential equation \frac{dy}{dx}=\frac{y^2-2x^2}{x^2} is homogeneous, since it is written in the standard form \frac{dy}{dx}=\frac{M(x,y)}{N(x,y)}, where M(x,y) and N(x,y) are the partial derivatives of a two-variable function f(x,y) and both are homogeneous functions of the same degree.
Final answer to the problem
$-\frac{1}{3}\ln\left(\frac{y}{x}+1\right)+\frac{1}{3}\ln\left(\frac{y}{x}-2\right)=\ln\left(x\right)+C_0$
