Separable equation
Separate variables so each side can be integrated.
Separable equation is one of 2 first order formulas in the differential equations section of this library, and it is used at university level.
Why separable equation works
If the right side splits into an x part times a y part, you can move all the y material to one side and all the x material to the other. Each side then depends on a single variable, so each can be integrated on its own. The chain rule is what licenses the move: integrating 1/h(y) times dy/dx with respect to x is the same as integrating 1/h(y) with respect to y.
What each symbol means
$g$ depends only on $x$ and $h$ only on $y$.
Separable equation: when it holds
Check solutions lost when dividing by $h(y)=0$ and apply the initial condition after integrating.
When it stops applying
Dividing by h(y) quietly deletes any constant solution where h(y) = 0. For y' = xy the general answer y = Ce^(x²/2) does cover y = 0 at C = 0, but in equations like y' = y² − 4 the constant solutions y = 2 and y = −2 must be checked and added back by hand.
Separable equation: a worked example
$y'=xy$ gives $\ln|y|=x^2/2+C$, so $y=Ce^{x^2/2}$.
The mistake to avoid
What people do: Students force an equation into separated form when the two variables are added rather than multiplied.
Why it goes wrong: Something like y' = x + y cannot be split, because no rearrangement puts x alone on one side and y alone on the other. Separation needs a product, not a sum.
Do this instead: Test whether the right side factors as g(x)h(y) before you start. If it does not, reach for the integrating factor method instead, which handles y' = x + y without complaint.
Separable equation: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $g$ depends only on $x$ and $h$ only on $y$.
- Check the conditions before substituting. Check solutions lost when dividing by $h(y)=0$ and apply the initial condition after integrating.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most differential equations slips.
Where this formula fits
- Subject
- Differential Equations formulas — 8 entries in this library
- Topic
- First order
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where separable equation comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Integration Techniques — the lesson behind this formula: choose substitution, parts, or simplification from structure.
- Calculus Calculator — check your substitution and the value it produces.
- Study differential equations — the subject guide that explains the ideas these formulas compress.
- Differential Equations Practice — questions that make you retrieve the formula instead of recognising it.
- All 8 differential equations formulas — the full grouped reference, or the complete formula library.
Questions about separable equation
Can I really treat dy/dx as a fraction here?
It is shorthand for a legitimate substitution, not a claim that the derivative is a fraction. The chain rule justifies every step, and the fraction notation is simply a fast way to write it.
Why is there only one constant of integration?
Both sides produce one, but subtracting them leaves a single unknown constant, so there is no point carrying two. Combining them early keeps the algebra clean.
Where does the absolute value in the logarithm go?
It gets absorbed into the constant. Solving ln|y| = x²/2 + C gives y = ±e^C e^(x²/2), and renaming ±e^C as a new constant C lets it take either sign.
How do I use the initial condition?
Integrate first, then substitute. Solving for C before you finish the integration is the most common way to get an answer that satisfies the starting point but not the equation.