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Separable Differential Equations
Separable Differential Equations
A first-order equation is separable if it can be written in the form $$\frac{dy}{dx}=g(x)h(y).$$
Method
Rewrite so all $y$ terms are with $dy$ and all $x$ terms are with $dx$, then integrate both sides.
Example 1
Solve $$\frac{dy}{dx}=2x.$$
Integrate: $$y=x^2+C.$$
Example 2
Solve $$\frac{dy}{dx}=xy.$$
Separate: $$\frac{1}{y}dy=x\,dx.$$ Integrate: $$\ln|y|=\frac{x^2}{2}+C,$$ so $$y=Ce^{x^2/2}.$$
Example 3
Solve $$y'=3y$$ with $y(0)=2$.
General solution: $y=Ce^{3x}$. Using $y(0)=2$ gives $C=2$, so $$y=2e^{3x}.$$
Practice Problems
1. Solve $y'=4x$.
2. Solve $y'=2y$.
3. What does "separable" mean?
4. Solve $y'=x^2y$ in general form.
5. If $y'=5y$ and $y(0)=1$, find $y$.
Show Answer Key
1. $y=2x^2+C$
2. $y=Ce^{2x}$
3. Variables can be rearranged with all $y$ terms on one side and all $x$ terms on the other
4. $y=Ce^{x^3/3}$
5. $y=e^{5x}$