Finding General Solutions: Separation of Variables - AP Calculus AB

1. What happens to the constant of integration when finding a particular solution?

   It is determined using an initial condition. Initial condition substitutes to find specific $C$ value.

2. Identify the error: $\frac{dy}{dx} = x^2y$ rewritten as $y \frac{dy}{dx} = x^2$.

   Correct: $\frac{1}{y} dy = x^2 dx$. Variables must be properly separated before integrating.

3. What is the role of the constant $C$ in the solution?

   It accounts for the family of solutions. Represents all possible curves in solution family.

4. What form must a differential equation have to use separation of variables?

   The form must be $N(y) \frac{dy}{dx} = M(x)$. Variables must be separable on opposite sides.

5. What is the general solution for $\frac{dy}{dx} = 0$?

   $y = C$, where $C$ is a constant. Zero derivative means $y$ is constant function.

6. What is the general solution of $\frac{dy}{dx} = ky$?

   $y = Ce^{kx}$, where $C$ is a constant. Standard exponential growth/decay model solution.

7. Separate variables and integrate: $\frac{dy}{dx} = \frac{2x}{y}$.

   $y dy = 2x dx$; integrate to $y^2 = x^2 + C$. Standard separation and integration process.

8. How do you verify a general solution found by separation of variables?

   Differentiate and substitute back into the original equation. Confirms the solution satisfies the original equation.

9. What is the result of integrating $e^{-y} dy = x dx$?

   $-e^{-y} = \frac{x^2}{2} + C$. Standard result from integrating separated variables.

10. What is the general solution of $\frac{dy}{dx} = \frac{x}{y}$ after separation of variables?

    $y^2 = x^2 + C$. Result from separating $y dy = x dx$ and integrating.

11. What is the solution of $\frac{dy}{dx} = y^2$ using separation of variables?

    $-\frac{1}{y} = x + C$. Separate: $y^{-2} dy = dx$, then integrate both sides.

12. What form does a differential equation take after separation?

    $N(y) dy = M(x) dx$. Standard separated form ready for integration.

13. What must you do after separating variables in a differential equation?

    Integrate both sides with respect to their variables. This yields the general solution with constant $C$.

14. What is the purpose of the constant $C$ in a general solution?

    It represents the constant of integration. Accounts for all possible solutions in the family.

15. How do you separate variables for $\frac{dy}{dx} = \frac{x}{\text{ln}(y)}$?

    Rewrite as $\text{ln}(y) dy = x dx$. Move $\ln(y)$ to denominator for proper separation.

16. State the form of a differential equation suitable for separation of variables.

    $N(y) dy = M(x) dx$ form. Variables separated on opposite sides of equation.

17. Find the error in separation: $y \frac{dy}{dx} = \frac{1}{x}$ rewritten as $y dy = \frac{1}{x} dx$.

    Correct form: $\frac{1}{y} dy = \frac{1}{x} dx$. Division by $y$ was missed in the separation.

18. State the integral of $y \frac{dy}{dx} = x$ after separation.

    $\frac{1}{2}y^2 = \frac{1}{2}x^2 + C$. Direct integration after variable separation.

19. How is the constant of integration $C$ determined in a particular solution?

    By using given initial conditions. Substitution yields specific solution from general form.

20. What is the next step after obtaining $y \frac{dy}{dx} = x$?

    Separate to $y dy = x dx$ and integrate both sides. Variables are now separated for integration.

21. In separation of variables, what is done after integrating?

    Solve for $y$ if possible to express $y$ explicitly. Convert implicit form to explicit if possible.

22. What is the solution to $\frac{dy}{dx} = y^2 x$?

    $-\frac{1}{y} = \frac{x^2}{2} + C$. Separate: $y^{-2} dy = x dx$, then integrate both sides.

23. What do you obtain after integrating both sides of a separated equation?

    An implicit solution, often in terms of $y$ and $x$. Integration produces this general form before solving for $y$.

24. What is the general solution of $\frac{dy}{dx} = \frac{1}{y}$?

    $y^2 = 2x + C$. Integrate $\int y dy = \int dx$ to get this result.

25. What is the first step in solving a differential equation using separation of variables?

    Rewrite the equation as $N(y) \frac{dy}{dx} = M(x)$. This isolates variables on separate sides for integration.

26. What is $y$ if $\frac{dy}{dx} = xy$ after integrating and solving?

    $y = Ce^{\frac{x^2}{2}}$. Exponential solution from integrating $\frac{dy}{y} = x dx$.

27. How do you solve $\frac{dy}{dx} = x e^{y}$ using separation of variables?

    Rewrite as $e^{-y} dy = x dx$ and integrate. Move exponential to left side before separating.

28. Identify the next step: $\frac{dy}{dx} = \frac{3x}{2y}$ rewritten as $2y dy = 3x dx$.

    Integrate both sides: $\frac{y^2}{2} = \frac{3x^2}{2} + C$. Standard integration after proper separation.

29. What does it mean if a solution is implicit?

    It is not solved explicitly for $y$ in terms of $x$. $y$ cannot be isolated algebraically from the equation.

30. Identify the integration step for $y \frac{dy}{dx} = x^2$.

    Integrate: $\frac{y^2}{2} = \frac{x^3}{3} + C$. Direct integration after separating variables properly.

31. What is the solution to $\frac{dy}{dx} = x^2 y^3$ after separation?

    $\frac{-1}{2y^2} = \frac{x^3}{3} + C$. Separate: $y^{-3} dy = x^2 dx$, then integrate.

32. What is the general solution of $\frac{dy}{dx} = \frac{1}{x}$?

    $y = \text{ln}|x| + C$. Direct integration of $\frac{1}{x}$ with respect to $x$.

33. Separate and integrate: $\frac{dy}{dx} = xy$.

    Rewrite as $\frac{1}{y} dy = x dx$ and integrate. Standard separation technique for this product form.

34. Find the general solution for $\frac{dy}{dx} = \frac{x}{y}$.

    $y^2 = x^2 + C$. Separate variables: $y dy = x dx$, then integrate.

35. What does the equation $dy = M(x) dx$ imply after separation?

    Integrate both sides to find $y = \text{integral of } M(x) dx + C$. Direct integration when variables are separated.

36. What is a key requirement for using separation of variables?

    The equation must be factored into $N(y) dy = M(x) dx$. Variables must be separable into this product form.

37. After separation and integration, what form does the solution take?

    An implicit equation involving $y$, $x$, and $C$. Variables not yet isolated to explicit $y$ form.

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