Solve the differential equation $y^{\prime}=\frac{y^2+x^2}{2xy}$

Step-by-step Solution

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Final answer to the problem

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

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equation
  • Homogeneous Differential Equation
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  • Product of Binomials with Common Term
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1

Rewrite the differential equation using Leibniz notation

Learn how to solve integrals by partial fraction expansion problems step by step online.

$\frac{dy}{dx}=\frac{y^2+x^2}{2xy}$

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Learn how to solve integrals by partial fraction expansion problems step by step online. Solve the differential equation y^'=(y^2+x^2)/(2xy). Rewrite the differential equation using Leibniz notation. We can identify that the differential equation \frac{dy}{dx}=\frac{y^2+x^2}{2xy} 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

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

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Function Plot

Plotting: $-\ln\left(\frac{y}{x}+1\right)-\ln\left(\frac{-y}{x}+1\right)=\ln\left(x\right)+C_0$

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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

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.

Used Formulas

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