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Exercise

$\frac{dz}{dx}=4xe^{3z-3x}$

Step-by-step Solution

1

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

$\frac{dz}{dx}=4xe^{3z}e^{-3x}$
2

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

$\frac{1}{e^{3z}}dz=4xe^{-3x}dx$
3

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

$\int\frac{1}{e^{3z}}dz=\int4xe^{-3x}dx$
4

Solve the integral $\int\frac{1}{e^{3z}}dz$ and replace the result in the differential equation

$\frac{-1}{3e^{3z}}=\int4xe^{-3x}dx$
5

Solve the integral $\int4xe^{-3x}dx$ and replace the result in the differential equation

$\frac{-1}{3e^{3z}}=\frac{-12x-4}{9e^{3x}}+C_0$

Final answer to the exercise

$\frac{-1}{3e^{3z}}=\frac{-12x-4}{9e^{3x}}+C_0$

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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