Modeling Situations with Differential Equations - AP Calculus AB

1. What is a solution to $y' = y$ with $y(0) = 2$?

   $y = 2e^x$. Exponential solution $y = Ce^x$ with $C = 2$ from condition.

2. Find the particular solution of $y' = 2x$ given $y(1) = 3$.

   $y = x^2 + 2$. Integrate $2x$ to get $x^2 + C$, then use initial condition.

3. Identify the particular solution of $y' = 2y$ given $y(1) = 3$.

   $y = 3e^{2x - 2}$. Apply exponential solution and initial condition at $x = 1$.

4. What is the solution to $y' = 3y$ with $y(0) = 5$?

   $y = 5e^{3x}$. Exponential solution with growth constant 3 and initial value 5.

5. What does it mean for a DE to be linear?

   Dependent variable and its derivatives are linear. No powers or products of $y$ or its derivatives.

6. Identify the independent variable in $dy/dt = -2y$.

   $t$. The variable in the denominator of the derivative.

7. State the general solution for $y' = \frac{1}{2}y$.

   $y = Ce^{\frac{1}{2}x}$. Exponential solution with growth rate $k = \frac{1}{2}$.

8. Identify the order of the differential equation $y'' + 3y' - 5y = 0$.

   Second order. The highest derivative is $y''$ (second derivative).

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

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

10. What is the integrating factor for $y' + py = q$?

    $e^{\int p \, dx}$. Multiplying factor that makes the left side an exact derivative.

11. What is an initial condition in differential equations?

    A value that specifies the solution of a DE at a point. Determines a unique solution from the general solution.

12. What does it mean for a DE to be separable?

    It can be written as $g(y)dy = f(x)dx$. Variables can be moved to opposite sides for integration.

13. What is the integrating factor for $y' + 3y = 0$?

    $e^{3x}$. For homogeneous equation, integrating factor is $e^{3x}$.

14. What is the order of the DE $x^2y'' + xy' + y = 0$?

    Second order. Highest derivative present is the second derivative $y''$.

15. Identify the solution to $y' = 4y$ with $y(0) = 1$.

    $y = e^{4x}$. Exponential solution $y = Ce^{4x}$ with $C = 1$.

16. Which method is used to solve $y' = yx$?

    Separation of variables. Variables can be separated: $\frac{dy}{y} = x dx$.

17. What method is used to solve $y' = xy$?

    Separation of variables. Can separate variables: $\frac{dy}{y} = x dx$.

18. What is the form of a linear first-order DE?

    $y' + p(x)y = q(x)$. Standard form for first-order linear differential equations.

19. What is the solution to $y' + y = 0$ with $y(0) = 3$?

    $y = 3e^{-x}$. Exponential decay with initial condition applied.

20. What is a first-order differential equation?

    An equation involving the first derivative of a function. Only involves $y'$, no higher derivatives.

21. What is a particular solution of a differential equation?

    A solution satisfying the initial conditions. General solution with constants determined by conditions.

22. What is a particular solution for $y' = xy$ given $y(1) = 4$?

    $y = 4e^{\frac{x^2 - 1}{2}}$. Separate variables and integrate, then apply condition.

23. Which function satisfies $y' = 0$?

    Constant functions. Zero derivative means no change, so $y$ is constant.

24. What is the general solution of $y' = 0$?

    Constant function. No change means the function remains constant.

25. What is the solution to $dy/dx = 3x^2$ with $y(0) = 1$?

    $y = x^3 + 1$. Integrate and apply the given initial condition.

26. What technique solves $y' + p(x)y = q(x)$?

    Integrating factor. Standard method for linear first-order equations.

27. What is the solution to $\frac{dy}{dx} = 3x^2$ with $y(0)=4$?

    $y = x^3 + 4$. Integrate $3x^2$ and apply the initial condition.

28. What is a homogeneous differential equation?

    All terms are a function of the dependent variable and its derivatives. No external forcing term, only function and derivatives.

29. What is a differential equation?

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

30. What is the general solution of $y' = ky$?

    $y = Ce^{kx}$. Exponential growth/decay solution form.

31. State the form of a separable differential equation.

    $\frac{dy}{dx} = g(x)h(y)$. Variables can be separated to each side of the equation.

32. Find the integrating factor for $y' + 2y = 3x$.

    $e^{2x}$. For $y' + py = q$, integrating factor is $e^{\int p dx} = e^{2x}$.

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