Final answer to the problem
$-\frac{1}{4}\ln\left|\frac{y}{x}+1\right|-\frac{3}{4}\ln\left|\frac{-y}{x}+1\right|=\ln\left|x\right|+C_0$
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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
Rewrite the differential equation using Leibniz notation
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$\frac{dy}{dx}=\frac{2x+y}{x+2y}$
Learn how to solve problems step by step online. Solve the differential equation y^'=(2x+y)/(x+2y). Rewrite the differential equation using Leibniz notation. We can identify that the differential equation \frac{dy}{dx}=\frac{2x+y}{x+2y} 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. Expand and simplify.
Final answer to the problem
$-\frac{1}{4}\ln\left|\frac{y}{x}+1\right|-\frac{3}{4}\ln\left|\frac{-y}{x}+1\right|=\ln\left|x\right|+C_0$
