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# Solve the differential equation $\frac{dy}{dx}=\frac{x}{2x-y}$

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##  Final answer to the problem

$-\ln\left|\frac{x}{y}-1\right|+\frac{y}{x-y}=\ln\left|y\right|+C_0$
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##  Step-by-step Solution 

How should I solve this problem?

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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

We can identify that the differential equation $\frac{dy}{dx}=\frac{x}{2x-y}$ 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

$\frac{dy}{dx}=\frac{x}{2x-y}$

Learn how to solve problems step by step online.

$\frac{dy}{dx}=\frac{x}{2x-y}$

Learn how to solve problems step by step online. Solve the differential equation dy/dx=x/(2x-y). We can identify that the differential equation \frac{dy}{dx}=\frac{x}{2x-y} 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: x=uy. Expand and simplify. Integrate both sides of the differential equation, the left side with respect to u, and the right side with respect to y.

##  Final answer to the problem

$-\ln\left|\frac{x}{y}-1\right|+\frac{y}{x-y}=\ln\left|y\right|+C_0$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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1
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3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch