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# Solve the differential equation $\left(x+1\right)\frac{dy}{dx}=x+6$

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##  Final answer to the problem

$y=x+1-\ln\left(x+1\right)+6\ln\left(x+1\right)+C_0$
Got another answer? Verify it here!

##  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

Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side of the equality

$dy=\frac{x+6}{x+1}dx$
2

Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$

$\int1dy=\int\frac{x+6}{x+1}dx$
3

Solve the integral $\int1dy$ and replace the result in the differential equation

$y=\int\frac{x+6}{x+1}dx$
4

Solve the integral $\int\frac{x+6}{x+1}dx$ and replace the result in the differential equation

$y=x+1-\ln\left(x+1\right)+6\ln\left(x+1\right)+C_0$

##  Final answer to the problem

$y=x+1-\ln\left(x+1\right)+6\ln\left(x+1\right)+C_0$

##  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

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: Differential Equations

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