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# Solve the differential equation $\frac{dy}{dx}=\frac{1}{\ln\left(2x+y+3\right)+1}-2$

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##  Final Answer

$\left(2x+y+3\right)\ln\left(2x+y+3\right)=x+C_0$
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##  Step-by-step Solution 

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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 $2x+y+3$ has the form $Ax+By+C$. Let's define a new variable $u$ and set it equal to the expression

$u=2x+y+3$

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$u=2x+y+3$

Learn how to solve problems step by step online. Solve the differential equation dy/dx=1/(ln(2x+y+3)+1)-2. 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 2x+y+3 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 2x+y+3 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

$\left(2x+y+3\right)\ln\left(2x+y+3\right)=x+C_0$

##  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

Linear Differential EquationExact Differential EquationSeparable Differential EquationHomogeneous Differential Equation

SnapXam A2

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1
2
3
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

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