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

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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 

How should I solve this problem?

• Choose an option
• Exact Differential Equation
• Linear Differential Equation
• Separable Differential Equation
• Homogeneous Differential Equation
• Integrate by partial fractions
• Product of Binomials with Common Term
• FOIL Method
• Integrate by substitution
• Integrate by parts
Can't find a method? Tell us so we can add it.
1

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.

$u=3x-y$

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

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

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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4
5
6
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

###  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.