To integrate \( y' = \left( \frac{e^t}{y} \right)^2 \), let's rewrite it in terms of differential variables and solve step-by-step.

Given:
\[
y' = \frac{dy}{dt} = \left( \frac{e^t}{y} \right)^2
\]

This can be rewritten as:
\[
\frac{dy}{dt} = \frac{e^{2t}}{y^2}
\]

To separate the variables \( y \) and \( t \), we can rewrite it as:
\[
y^2 \, dy = e^{2t} \, dt
\]

Now, integrate both sides:

1. **Integrate the left side:**
   \[
   \int y^2 \, dy = \frac{y^3}{3} + C_1
   \]

2. **Integrate the right side:**
   \[
   \int e^{2t} \, dt = \frac{e^{2t}}{2} + C_2
   \]

So, we have:
\[
\frac{y^3}{3} = \frac{e^{2t}}{2} + C
\]

where \( C = C_2 - C_1 \), a constant.

Finally, multiply both sides by 3 to isolate \( y^3 \):
\[
y^3 = \frac{3}{2} e^{2t} + 3C
\]

Taking the cube root of both sides, we find:
\[
y = \sqrt[3]{\frac{3}{2} e^{2t} + 3C}
\]

This is the solution for \( y \) in terms of \( t \).

Would you like further details or have any questions?

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### Related Questions

1. How do you solve a first-order differential equation by separation of variables?
2. What is the general approach to integrating functions of the form \( y' = f(t, y) \)?
3. How can constant factors be handled during integration in differential equations?
4. How does the method of separation of variables apply to other differential equations?
5. What are the conditions for the existence and uniqueness of solutions to differential equations?

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### Tip

For differential equations, always check if separating variables is possible as a first step—it simplifies the problem significantly.

This solution guides you through integrating the differential equation y' = (e^t/y)^2 using the separation of variables method. Learn to manipulate and solve differential equations with exponential functions, step-by-step, ideal for undergraduate calculus students.

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