Solve a separable differential equation and plot solution families

Primary Tool

2D Graphing

The plotter is the best primary tool because the teaching point is not only the symbolic answer $y = C e^{x^{2} /2}$ , but how that one formula becomes many curves.

Problem

Solve $y^{'} = x y$ and describe the family of solutions.

This is a standard example because it shows what a separable equation is supposed to look like before more complicated methods appear.

From algebra to geometry

Separating variables gives $\frac{1}{y} d y = x d x$ . After integration you get $ln ∣ y ∣ = x^{2} /2 + C$ , which becomes $y = C e^{x^{2} /2}$ after absorbing sign information into the constant.

The graph matters because it shows that the constant is not a small correction. Changing $C$ changes the entire curve and the sign of the solution.

Step-by-step walkthrough

The symbolic solution and the plotted family should tell the same story, so it helps to write the method in explicit stages.

1Start from $y^{'} = x y$ and rewrite it as $\frac{d y}{d x} = x y$ .
2Separate variables by dividing by $y$ and multiplying by $d x$ , giving $\frac{1}{y} d y = x d x$ .
3Integrate both sides to obtain $ln ∣ y ∣ = x^{2} /2 + C$ .
4Exponentiate to solve for $y$ , giving $∣ y ∣ = e^{x^{2} /2 + C} = e^{C} e^{x^{2} /2}$ .
5Absorb the constant and sign into one parameter $C$ to write the family as $y = C e^{x^{2} /2}$ .
6Use the graph to compare several values of $C$ so the family behavior is visible, not just symbolic.

Common Pitfall

Do not stop at $ln ∣ y ∣ = x^{2} /2 + C$ . You still need to solve for $y$ so the whole family of solutions is written explicitly.

Try a Variation

Use the same workflow on $y^{'} = 2 x y$ . Which part changes, and what happens to the exponent in the solution family?

Keep moving through the cluster

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ProofReal AnalysisProof Builder

Epsilon-delta proof that $lim_{x \to 2} x^{2} = 4$

This proof is common because it shows the standard move in early analysis: factor the expression, then bound the extra factor by forcing $x$ into a smaller interval first.

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ExplanationCalculus2D Graphing

What a derivative means geometrically

The derivative is the slope of the tangent line, but that phrase only becomes useful when you compare the curve to nearby secant lines and local linear approximations.

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