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

Step-by-step Solution

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Final answer to the problem

$y=\ln\left(C_0-\cos\left(x\right)\right)$
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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
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  • Integrate by parts
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Rewrite the differential equation in the standard form $M(x,y)dx+N(x,y)dy=0$

$e^ydy-\sin\left(x\right)dx=0$

Learn how to solve integral calculus problems step by step online.

$e^ydy-\sin\left(x\right)dx=0$

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Learn how to solve integral calculus problems step by step online. Solve the differential equation dy/dx=sin(x)/(e^y). Rewrite the differential equation in the standard form M(x,y)dx+N(x,y)dy=0. The differential equation e^ydy-\sin\left(x\right)dx=0 is exact, since it is written in the standard form M(x,y)dx+N(x,y)dy=0, where M(x,y) and N(x,y) are the partial derivatives of a two-variable function f(x,y) and they satisfy the test for exactness: \displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form f(x,y)=C. Using the test for exactness, we check that the differential equation is exact. Integrate M(x,y) with respect to x to get.

Final answer to the problem

$y=\ln\left(C_0-\cos\left(x\right)\right)$

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Function Plot

Plotting: $\frac{dy}{dx}+\frac{-\sin\left(x\right)}{e^y}$

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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
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: Integral Calculus

Integration assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Used Formulas

See formulas (3)

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