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Solve the differential equation $3\left(1+f^2\right)\left(\frac{dy}{df}\right)=y\left(y^3-1\right)$

Step-by-step Solution

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Solution

Final answer to the problem

$3\ln\left|y\right|-\ln\left|y+1\right|-\ln\left|y^{2}-y+1\right|=\arctan\left(f\right)+C_0$
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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 $f$ variable to the right side of the equality

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$\frac{3}{y\left(y^3-1\right)}dy=\frac{1}{1+f^2}df$

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Learn how to solve problems step by step online. Solve the differential equation 3(1+f^2)dy/df=y(y^3-1). Group the terms of the differential equation. Move the terms of the y variable to the left side, and the terms of the f variable to the right side of the equality. Simplify the expression \frac{3}{y\left(y^3-1\right)}dy. Integrate both sides of the differential equation, the left side with respect to y, and the right side with respect to f. Solve the integral \int\frac{3}{y\left(y+1\right)\left(y^{2}-y+1\right)}dy and replace the result in the differential equation.

Final answer to the problem

$3\ln\left|y\right|-\ln\left|y+1\right|-\ln\left|y^{2}-y+1\right|=\arctan\left(f\right)+C_0$

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Function Plot

Plotting: $3\left(1+f^2\right)\left(\frac{dy}{df}\right)-y\left(y^3-1\right)$

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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
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

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