Solve the given differential equation by separation of variables: $\frac{dy}{dx} = e^{3x + 5y}$

Neetesh Kumar | November 08, 2024

Differential Equation Homework Help

This is the solution to Math 2A, section 13Z, Fall 2023 | WebAssign
Math002ACh2Sec02 (Homework) Question - 1
Contact me if you need help with Homework, Assignments, Tutoring Sessions, or Exams for STEM subjects.
You can see our Testimonials or Vouches from here of the previous works I have done.

Get Homework Help

---

Step-by-Step-Solution:

To solve this equation, we first rearrange it to separate the variables $y$ and $x$:

$\frac{dy}{dx} = e^{3x} e^{5y}$

Now, we can separate the variables:

$\frac{dy}{e^{5y}} = e^{3x} \, dx$

Step 1: Integrate Both Sides

Next, we integrate both sides:

$\int \frac{dy}{e^{5y}} = \int e^{3x} \, dx$

The left-hand side can be rewritten as:

$\int e^{-5y} \, dy = \frac{1}{-5} e^{-5y} + C_1$

The right-hand side integrates to:

$\int e^{3x} \, dx = \frac{1}{3} e^{3x} + C_2$

Combining these results gives:

$\frac{1}{-5} e^{-5y} = \frac{1}{3} e^{3x} + C$

Step 2: Solve for ( y )

Now, we can solve for ( y ):

Multiplying through by -5:

$e^{-5y} = -\frac{5}{3} e^{3x} - 5C$

Taking the natural logarithm of both sides gives:

$-5y = \ln\left(-\frac{5}{3} e^{3x} - 5C\right)$

Therefore:

$y = -\frac{1}{5} \ln\left(-\frac{5}{3} e^{3x} - 5C\right)$

Conclusion

The general solution is:

$\boxed{e^{-5y} = -\frac{5e^{3x}}{3} + C}$

---

Please comment below if you find any error in this solution.
If this solution helps, then please share this with your friends.
Please subscribe to my Youtube channel for video solutions to similar questions.
Keep Smiling :-)