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

Step-by-step Solution

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Final Answer

$y=\frac{e^x\sin\left(x\right)-e^x\cos\left(x\right)}{2e^x}$
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Step-by-step Solution

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We can identify that the differential equation has the form: $\frac{dy}{dx} + P(x)\cdot y(x) = Q(x)$, so we can classify it as a linear first order differential equation, where $P(x)=1$ and $Q(x)=\sin\left(x\right)$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$

$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$

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$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$

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Learn how to solve differential equations problems step by step online. Solve the differential equation dy/dx+y=sin(x). We can identify that the differential equation has the form: \frac{dy}{dx} + P(x)\cdot y(x) = Q(x), so we can classify it as a linear first order differential equation, where P(x)=1 and Q(x)=\sin\left(x\right). In order to solve the differential equation, the first step is to find the integrating factor \mu(x). To find \mu(x), we first need to calculate \int P(x)dx. So the integrating factor \mu(x) is. Now, multiply all the terms in the differential equation by the integrating factor \mu(x) and check if we can simplify.

Final Answer

$y=\frac{e^x\sin\left(x\right)-e^x\cos\left(x\right)}{2e^x}$

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Function Plot

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

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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
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: Differential Equations

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

Used Formulas

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