Finding General Solutions Using Separation of Variables AB & BC
An equation separates when its right side factors as a function of $x$ times a function of $y$: $xy$, $y\cos x$ and $\frac{x + 1}{y^{2}}$ all do, while $x + y$ and $\sin(x + y)$ do not, and $xy + x$ does because it factors as $x(y + 1)$. Separating means every $y$ on the side with $dy$ and every $x$ on the side with $dx$; a stray $y$ left on the $dx$ side means integrating is not yet legal.
Integrating both sides introduces exactly ONE constant, and where it ends up is the whole difficulty. From $\frac{dy}{dx} = xy$ comes $\ln|y| = \frac{x^{2}}{2} + C$, and exponentiating turns that into $|y| = e^{C}e^{x^{2}/2}$, so the constant becomes a FACTOR: $y = Ce^{x^{2}/2}$, not $e^{x^{2}/2} + C$. Those two families are stretches and shifts of the same curve and are not interchangeable. Other separable forms leave the constant elsewhere, as in $y^{2} = 2x^{2} + C$ or $y = -\frac{1}{x + C}$, and an implicit answer is a complete general solution.
The work
3 ways in · any order
Lesson
Finding General Solutions Using Separation of Variables
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Tests whether an equation separates by trying to factor its right side, then runs one example from separation through integration to the solved form with the constant tracked at every line, showing why it becomes a multiplicative factor after exponentiating and covering the power, reciprocal and logarithmic endings.
Diagnostic
10-item topic check
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Ten items on separation of variables: recognising which equations separate at all, completing the separation before integrating, and placing the single constant of integration where the algebra puts it rather than adding it to the finished answer.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.