Modeling Situations with Differential Equations - AP Calculus BC

1. Identify the type of differential equation: $y'' + y = \tan(x)$.

   Second-order non-homogeneous differential equation. Second-order with non-zero right side.

2. Identify the independent variable in $dz/dt = z^2 + t^2$.

   The independent variable is $t$. Variable with respect to which we differentiate.

3. What is a differential equation?

   An equation involving derivatives of a function. Relates a function to its rate of change.

4. Solve $y' = \frac{1}{x}$ for $y$.

   $y = \text{ln}|x| + C$. Antiderivative of $\frac{1}{x}$ is $\ln|x|$.

5. Identify the dependent variable in $dz/dt = z^2 + t^2$.

   The dependent variable is $z$. Function being differentiated ($z$ depends on $t$).

6. Solve the differential equation $\frac{dy}{dx} = 3x^2$.

   $y = x^3 + C$. Direct integration: $\int 3x^2 dx = x^3 + C$.

7. Identify the homogeneous part of $y'' + 5y' + 6y = e^x$.

   $y'' + 5y' + 6y$. Left side when right side equals zero.

8. What is the characteristic equation of $y'' + 2y' + 5y = 0$?

   $r^2 + 2r + 5 = 0$. Substitute $y'' = r^2$, $y' = r$ into equation.

9. Solve for $y$ in the differential equation $\frac{dy}{dx} = 0$.

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

10. What function solves $y' - 2y = 0$?

    $y = Ce^{2x}$. Exponential solution with coefficient $k = 2$.

11. What type of differential equation is $y' + p(x)y = g(x)$?

    First-order linear differential equation. Standard form with $y$ and $y'$ terms only.

12. What is the solution for $dy/dx = y$ using separation of variables?

    $y = Ce^{x}$. Separate variables: $\frac{dy}{y} = dx$, integrate.

13. What does it mean for a function to be a solution to a differential equation?

    Substituting it into the equation yields a true statement. The function satisfies the equation identically.

14. Identify the dependent variable in $dy/dx = x^2 + y^2$.

    The dependent variable is $y$. Variable being differentiated ($y$ depends on $x$).

15. Define a separable differential equation.

    An equation where variables can be separated on opposite sides. Variables can be moved to opposite sides.

16. Define a non-homogeneous differential equation.

    An equation with non-zero terms not involving the dependent variable. Contains terms independent of dependent variable.

17. What is the characteristic equation of $y'' + 4y' + 4y = 0$?

    $r^2 + 4r + 4 = 0$. Replace $y''$ with $r^2$, $y'$ with $r$.

18. What is the general solution to $y'' + 4y = 0$?

    $y = C_1 \text{cos}(2x) + C_2 \text{sin}(2x)$. Characteristic equation $r^2 + 4 = 0$ gives $r = ±2i$.

19. Solve for $y$ in $dy/dx = 2xy$ using separation of variables.

    $y = Ce^{x^2}$. Separate: $\frac{dy}{y} = 2x dx$, integrate both sides.

20. What is a homogeneous differential equation?

    An equation where $0$ is a solution when all terms set to $0$. All terms involve the dependent variable.

21. What is the solution to $y'' - y = 0$?

    $y = C_1e^x + C_2e^{-x}$. Characteristic equation $r^2 - 1 = 0$ gives $r = ±1$.

22. Identify the linear part of $y'' + 4y' + 4y = 5x$.

    $y'' + 4y' + 4y$. Terms involving $y$ and its derivatives.

23. Identify the order of the differential equation $y'' + 3y' - 4y = 0$.

    Order is 2. Highest derivative is second order ($y''$).

24. What is the order of the equation $d^3y/dx^3 + y = 0$?

    Order is 3. Highest derivative is third order.

25. What is the integrating factor for $y' + p(x)y = g(x)$?

    $e^{\int p(x) \text{d}x}$. Multiplier that makes equation exact.

26. What is the integrating factor for $dy/dx + 2y = e^x$?

    $e^{2x}$. Exponential of integral of coefficient: $e^{\int 2 dx}$.

27. Identify the independent variable in $dy/dx = x^2 + y^2$.

    The independent variable is $x$. Variable with respect to which we differentiate.

28. State the general solution form for $y' = ky$.

    $y = Ce^{kt}$, where $C$ is a constant. Exponential growth/decay model solution.

29. What is the general solution to $y'' + 9y = 0$?

    $y = C_1 \text{cos}(3x) + C_2 \text{sin}(3x)$. Characteristic equation $r^2 + 9 = 0$ gives $r = ±3i$.

30. Find the derivative of $y = Ce^{kt}$.

    $y' = kCe^{kt}$. Apply chain rule: $\frac{d}{dt}(Ce^{kt}) = kCe^{kt}$.

31. State a real-world example of a first-order differential equation.

    Newton's Law of Cooling. Temperature change proportional to difference.

32. What is a particular solution?

    A solution satisfying both the differential equation and initial conditions. Specific solution meeting initial conditions.

33. What method is used to solve $y' = ky$?

    Separation of variables. Rearrange to separate $y$ and $x$ terms.

34. Define a linear differential equation.

    An equation where the dependent variable and derivatives appear linearly. Dependent variable appears to first power only.

35. Write the differential equation for exponential growth.

    $\frac{dy}{dt} = ky$. Rate proportional to current amount.

36. What is the order of $d^2y/dx^2 + 3y' + 4y = x^2$?

    Order is 2. Highest derivative is second order.

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