Therefore $A = 2$, giving us the particular solution:
$$\boxed{y = 2\,e^{x^2/2}}$$
Step 6 — Equilibrium Check
Here $g(y) = y$, so $y = 0$ is an equilibrium solution.
Our initial condition $y(0) = 2 \neq 0$, so the equilibrium doesn't apply here.
The solution $y = 2e^{x^2/2}$ is defined for all real $x$ (global existence).
Common Pitfall: Lost Solutions
Critical Warning
When we divide by $g(y)$, we assume $g(y) \neq 0$.
The values where $g(y) = 0$ give equilibrium (constant) solutions that can be lost during separation!
Example: Consider $\dfrac{dy}{dx} = y(1-y)$
What separation gives
$$\int \frac{dy}{y(1-y)} = \int dx$$
General family of logistic curves — but misses the constant solutions!
The lost solutions
Setting $g(y) = y(1-y) = 0$:
$$y = 0 \quad \text{and} \quad y = 1$$
Always check these separately!
Implicit vs. Explicit Solutions
Sometimes we can't solve for $y$ explicitly — and that's perfectly fine.
Explicit Solution
$y$ written directly as a function of $x$
$$y = 2e^{x^2/2}$$
Plug in any $x$ and get $y$ directly.
Implicit Solution
An equation relating $x$ and $y$
$$y^2 + \sin(y) = x^3 + C$$
Can't isolate $y$ — but still a valid solution!
Verification tip: For implicit solutions, use implicit differentiation to verify. Differentiate both sides with respect to $x$, and confirm you recover the original ODE.
Key Takeaways
🎯
Recognize separable form
$\frac{dy}{dx} = f(x) \cdot g(y)$ — the RHS must be a product of a function of $x$ and a function of $y$.
✂️
Separate, then integrate
Move all $y$'s to one side, all $x$'s to the other, and integrate. Don't forget the $+C$!
⚠️
Never lose equilibrium solutions
When dividing by $g(y)$, check $g(y) = 0$ separately. These constant solutions are easy to miss!
✅
Verify your answer
Substitute back into the original ODE. For implicit solutions, use implicit differentiation.