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Solve the differential equation $6y\left(\frac{dy}{dx}\right)=\frac{4x\cos\left(x\right)}{\sin\left(y\right)^2}$

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Solution

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Final answer to the problem

$6\left(y\left(\frac{1}{2}y-\frac{1}{4}\sin\left(2y\right)\right)-\frac{1}{4}y^2-\frac{1}{8}\cos\left(2y\right)\right)=4x\sin\left(x\right)+4\cos\left(x\right)+C_0$
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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
  • 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

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

$6y\sin\left(y\right)^2dy=4x\cos\left(x\right)dx$

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Learn how to solve differential equations problems step by step online. Solve the differential equation 6ydy/dx=(4xcos(x))/(sin(y)^2). 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 6y\sin\left(y\right)^2dy. 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 \int6y\left(1-\cos\left(y\right)^2\right)dy and replace the result in the differential equation.

Final answer to the problem

$6\left(y\left(\frac{1}{2}y-\frac{1}{4}\sin\left(2y\right)\right)-\frac{1}{4}y^2-\frac{1}{8}\cos\left(2y\right)\right)=4x\sin\left(x\right)+4\cos\left(x\right)+C_0$

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Function Plot

Plotting: $6y\left(\frac{dy}{dx}\right)+\frac{-4x\cos\left(x\right)}{\sin\left(y\right)^2}$

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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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Similar Problems Solved

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

Main Topic: Differential Equations

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

Used Formulas

See formulas (9)

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