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Exercise

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

Step-by-step Solution

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 $y$

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

Final answer to the exercise

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

Try other ways to solve this exercise

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equations
  • 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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