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

Step-by-step Solution

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Solution

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Final answer to the problem

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

Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side of the equality

$\frac{1}{y^2-y}dy=\frac{1}{x}dx$

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

$\frac{1}{y^2-y}dy=\frac{1}{x}dx$

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Learn how to solve differential equations problems step by step online. Solve the differential equation xdy/dx=y^2-y. Group the terms of the differential equation. Move the terms of the y variable to the left side, and the terms of the x variable to the right side of the equality. Simplify the expression \frac{1}{y^2-y}dy. Integrate both sides of the differential equation, the left side with respect to y, and the right side with respect to x. Solve the integral \int\frac{1}{y\left(y-1\right)}dy and replace the result in the differential equation.

Final answer to the problem

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

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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

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Function Plot

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

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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
D□x
∫
∫◻
|◻|
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:

Similar Problems Solved

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

$\frac{dy}{dx}=y\left(y+2\right)$

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

Main Topic: Differential Equations

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

Used Formulas

See formulas (3)

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