Separable differential equations
Problem 6.58 · medium
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 x y,\ y(0) = 3 \).
- Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
- Use y(0) to find the constant of integration, then solve for y.
- \[ \frac{d}{d x} 3 e^{- \frac{3 x^{2}}{2}} = - 9 x e^{- \frac{3 x^{2}}{2}} \]The solution satisfies the equation.✓ Proved
- \[ 3 \]And the initial condition.✓ Proved
Answer \( y = 3 e^{- \frac{3 x^{2}}{2}} \)
Lines: 2 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | sympy.checkodesol substitutes the solution into the equation; y(0) matches |
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/separable_ode, checked 2026-09-26 with SymPy 1.14.0.