Exponential growth/decay
\[\frac{dP}{dt}=kP\;\Rightarrow\;P=P_0e^{kt}\]
Differential Equations
Variables Separable Method
Understand
If \(\frac{dy}{dx}=f(x)\,g(y)\), separate the variables and integrate both sides:
\[\int\frac{dy}{g(y)}=\int f(x)\,dx+C.\]
Example: \(\frac{dy}{dx}=\frac{x}{y}\Rightarrow y\,dy=x\,dx\Rightarrow\frac{y^2}{2}=\frac{x^2}{2}+C\), i.e. \(y^2-x^2=2C\).
Growth and decay. \(\frac{dP}{dt}=kP\Rightarrow\frac{dP}{P}=k\,dt\Rightarrow P=P_0e^{kt}\). This models population growth (\(k>0\)), radioactive decay and cooling (\(k<0\)).
Key Concepts
- Get all y-terms with dy and x-terms with dx.
- Integrate both sides; one constant is enough.
- Use the initial condition to fix C.
Formula Bank
Common Mistakes
Losing the constant
After \(\log|y|=x^2+C\), write \(y=Ae^{x^2}\) – the constant multiplies, it does not add.
Practice & Topic Test
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Separate, integrate, add one constant, apply the condition.
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