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

## Step-by-step Solution

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###  Videos

$y=\frac{-\ln\left(-\frac{2}{3}e^{3x}+C_1\right)}{2}$
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##  Step-by-step Solution 

Specify the solving method

1

Apply the property of the product of two powers of the same base in reverse: $a^{m+n}=a^m\cdot a^n$

$\frac{dy}{dx}=e^{3x}e^{2y}$
2

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

$\frac{1}{e^{2y}}dy=e^{3x}dx$
3

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

$\int\frac{1}{e^{2y}}dy=\int e^{3x}dx$
4

Solve the integral $\int\frac{1}{e^{2y}}dy$ and replace the result in the differential equation

$\frac{-1}{2e^{2y}}=\int e^{3x}dx$
5

Solve the integral $\int e^{3x}dx$ and replace the result in the differential equation

$\frac{-1}{2e^{2y}}=\frac{1}{3}e^{3x}+C_0$
6

Find the explicit solution to the differential equation. We need to isolate the variable $y$

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

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

##  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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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.