Step-by-step Solution

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

Go!
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Final Answer

$y=\frac{\ln\left(\frac{-\frac{1}{2}}{\frac{1}{3}e^{3x}+C_0}\right)}{2}$

Step-by-step explanation

Problem to solve:

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

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

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

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

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

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{-\frac{1}{2}}{e^{2y}}=\frac{1}{3}e^{3x}+C_0$
7

Take the reciprocal of both sides of the equation

$\frac{e^{2y}}{-\frac{1}{2}}=\frac{1}{\frac{1}{3}e^{3x}+C_0}$
8

Multiply both sides of the equation by $-\frac{1}{2}$

$e^{2y}=-\frac{1}{2}\left(\frac{1}{\frac{1}{3}e^{3x}+C_0}\right)$
9

Multiply the fraction and term

$e^{2y}=\frac{-\frac{1}{2}}{\frac{1}{3}e^{3x}+C_0}$
10

We can take out the unknown from the exponent by applying natural logarithm to both sides of the equation

$\ln\left(e^{2y}\right)=\ln\left(\frac{-\frac{1}{2}}{\frac{1}{3}e^{3x}+C_0}\right)$
11

Apply the formula: $\ln\left(e^x\right)$$=x$, where $x=2y$

$2y=\ln\left(\frac{-\frac{1}{2}}{\frac{1}{3}e^{3x}+C_0}\right)$
12

Divide both sides of the equation by $2$

$y=\frac{\ln\left(\frac{-\frac{1}{2}}{\frac{1}{3}e^{3x}+C_0}\right)}{2}$

Final Answer

$y=\frac{\ln\left(\frac{-\frac{1}{2}}{\frac{1}{3}e^{3x}+C_0}\right)}{2}$