Unit 7: Differential Equations
1. Separation of variables (free preview)
A differential equation is separable when it can be written as a product of an x-part and a y-part. The method: get all $y$ terms (with $dy$) on one side, all $x$ terms (with $dx$) on the other, integrate both sides, then use the initial condition to pin down $C$.
Worked example
Solve $\dfrac{dy}{dx} = \dfrac{x}{y}$ with $y(0) = 2$.
- Separate: $y\,dy = x\,dx$.
- Integrate: $\dfrac{y^2}{2} = \dfrac{x^2}{2} + C$.
- Apply $y(0) = 2$: $2 = C$, so $y^2 = x^2 + 4$.
- Solve for $y$, keeping the branch matching the initial condition: $y = \sqrt{x^2 + 4}$.
Common mistake: solving for $C$ before integrating, or forgetting $C$ entirely. On the FRQ, the constant appears immediately after integrating, and the initial condition is applied to that equation, not to the separated form.
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The rest of Unit 7 is in the full course
Short unit, reliable FRQ appearance, very learnable points:
- Slope fields: reading them, matching them to equations
- Exponential growth and decay: $\tfrac{dy}{dt} = ky$ from words to solution
- Equilibrium solutions and domain-of-solution traps
- Initial value problems, start to finish
- 35 exam-style questions with verified step-by-step solutions
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