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Solve the differential equation $\frac{dy}{dx}=\sin\left(3x-y\right)$

Step-by-step Solution

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Final answer to the problem

$\ln\left(-2\tan\left(\frac{3x-y}{2}\right)+3\right)=-x+C_1$
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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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When we identify that a differential equation has an expression of the form $Ax+By+C$, we can apply a linear substitution in order to simplify it to a separable equation. We can identify that $3x-y$ has the form $Ax+By+C$. Let's define a new variable $u$ and set it equal to the expression

$u=3x-y$

Learn how to solve integrals of exponential functions problems step by step online.

$u=3x-y$

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Learn how to solve integrals of exponential functions problems step by step online. Solve the differential equation dy/dx=sin(3x-y). When we identify that a differential equation has an expression of the form Ax+By+C, we can apply a linear substitution in order to simplify it to a separable equation. We can identify that 3x-y has the form Ax+By+C. Let's define a new variable u and set it equal to the expression. Isolate the dependent variable y. Differentiate both sides of the equation with respect to the independent variable x. Now, substitute 3x-y and \frac{dy}{dx} on the original differential equation. We will see that it results in a separable equation that we can easily solve.

Final answer to the problem

$\ln\left(-2\tan\left(\frac{3x-y}{2}\right)+3\right)=-x+C_1$

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Function Plot

Plotting: $\frac{dy}{dx}-\sin\left(3x-y\right)$

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5
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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: Integrals of Exponential Functions

Those are integrals that involve exponential functions. Recall that an exponential function is a function of the form f(x)=a^x.

Used Formulas

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