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

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Final answer to the problem

$y=x\tan\left(\ln\left(x\right)+C_0\right)$
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Step-by-step Solution

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

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

Learn how to solve differential equations problems step by step online.

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

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Learn how to solve differential equations problems step by step online. Solve the differential equation dy/dx=(x^2+xyy^2)/(x^2). We can identify that the differential equation \frac{dy}{dx}=\frac{x^2+xy+y^2}{x^2} 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. Group the terms of the differential equation. Move the terms of the u variable to the left side, and the terms of the x variable to the right side of the equality.

Final answer to the problem

$y=x\tan\left(\ln\left(x\right)+C_0\right)$

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Function Plot

Plotting: $\frac{dy}{dx}+\frac{-x^2-xy-y^2}{x^2}$

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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: Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives.

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