Final answer to the problem
$\frac{x}{6y}-\frac{1}{2}\ln\left|\frac{y}{x}\right|=\ln\left|x\right|+C_0$
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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
Cancel $8$ from both sides of the equation
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$x\left(x+3y\right)\frac{dy}{dx}=y\left(x-3y\right)$
Learn how to solve problems step by step online. Solve the differential equation 8x(x+3y)dy/dx=8y(x-3y). Cancel 8 from both sides of the equation. Rewrite the differential equation. We can identify that the differential equation \frac{dy}{dx}=\frac{y\left(x-3y\right)}{x\left(x+3y\right)} 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. Use the substitution: y=ux.
Final answer to the problem
$\frac{x}{6y}-\frac{1}{2}\ln\left|\frac{y}{x}\right|=\ln\left|x\right|+C_0$
