Solve the differential equation $2x\cdot dx=\left(x+y\right)dy$

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Final answer to the problem

$-\frac{2}{3}\ln\left|\frac{-x}{y}+1\right|-\frac{1}{3}\ln\left|\frac{2x}{y}+1\right|=\ln\left|y\right|+C_0$
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We can identify that the differential equation $2x\cdot dx=\left(x+y\right)dy$ is homogeneous, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, 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

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$2x\cdot dx=\left(x+y\right)dy$

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Learn how to solve integrals by partial fraction expansion problems step by step online. Solve the differential equation 2xdx=(x+y)dy. We can identify that the differential equation 2x\cdot dx=\left(x+y\right)dy is homogeneous, since it is written in the standard form M(x,y)dx+N(x,y)dy=0, 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

$-\frac{2}{3}\ln\left|\frac{-x}{y}+1\right|-\frac{1}{3}\ln\left|\frac{2x}{y}+1\right|=\ln\left|y\right|+C_0$

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Function Plot

Plotting: $2x\cdot dx-\left(x+y\right)dy$

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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

How to improve your answer:

Main Topic: Integrals by Partial Fraction Expansion

The partial fraction decomposition or partial fraction expansion of a rational function is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

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