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

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Solution

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Final answer to the problem

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

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  • 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
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1

Rewrite the differential equation using Leibniz notation

Learn how to solve integration by substitution problems step by step online.

$\frac{dy}{dx}=e^{\left(2x+3y\right)}$

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Unlock the first 3 steps of this solution

Learn how to solve integration by substitution problems step by step online. Solve the differential equation y^'=e^(2x+3y). Rewrite the differential equation using Leibniz notation. Apply the property of the product of two powers of the same base in reverse: a^{m+n}=a^m\cdot a^n. 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. Integrate both sides of the differential equation, the left side with respect to y, and the right side with respect to x.

Final answer to the problem

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

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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

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Function Plot

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

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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
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:

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Main Topic: Integration by Substitution

In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative.

Used Formulas

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