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

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Solution

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Final answer to the problem

$y=\sqrt{2\left(\frac{x^{7}}{7}+\frac{3x^{5}}{5}+x^{3}+x+C_0\right)},\:y=-\sqrt{2\left(\frac{x^{7}}{7}+\frac{3x^{5}}{5}+x^{3}+x+C_0\right)}$
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Step-by-step Solution

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  • Exact Differential Equation
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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

$y\cdot dy=\left(x^2+1\right)^3dx$

Learn how to solve logarithmic differentiation problems step by step online.

$y\cdot dy=\left(x^2+1\right)^3dx$

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Learn how to solve logarithmic differentiation problems step by step online. Solve the differential equation dy/dxy=(x^2+1)^3. 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 \left(x^2+1\right)^3dx. Integrate both sides of the differential equation, the left side with respect to y, and the right side with respect to x. Expand the integral \int\left(x^{6}+3x^{4}+3x^2+1\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately.

Final answer to the problem

$y=\sqrt{2\left(\frac{x^{7}}{7}+\frac{3x^{5}}{5}+x^{3}+x+C_0\right)},\:y=-\sqrt{2\left(\frac{x^{7}}{7}+\frac{3x^{5}}{5}+x^{3}+x+C_0\right)}$

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Function Plot

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

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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

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Main Topic: Logarithmic Differentiation

The logarithmic derivative of a function f(x) is defined by the formula f'(x)/f(x).

Used Formulas

See formulas (6)

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