Verifying Solutions for Differential Equations - AP Calculus AB

1. What is a particular solution?

   A solution with specified initial conditions. General solution evaluated at given conditions.

2. What is a separable differential equation?

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

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

   Order 2. Highest derivative is the second, so order is 2.

4. What is the general solution for $y' = -y$?

   $y = Ce^{-x}$. Exponential decay with rate constant 1.

5. Verify if $y = C \text{cos}(2x)$ solves $y'' + 4y = 0$.

   Yes, it satisfies $y'' + 4y = 0$. $y'' = -4C\cos(2x)$, so $y'' + 4y = 0$.

6. Verify if $y = \frac{1}{2}e^{2x}$ solves $y' = y$.

   No, $y' = e^{2x} \neq y$. $y' = e^{2x} \neq \frac{1}{2}e^{2x} = y$.

7. What is the general solution for $y'' = 0$?

   $y = C_1x + C_2$. Double integration of zero gives linear function.

8. What is a differential equation?

   An equation involving derivatives of a function. Mathematical relationship involving rates of change.

9. What is the general solution for $y' = 2y$?

   $y = Ce^{2x}$. Exponential growth with rate constant 2.

10. What is a linear differential equation?

    An equation of the form $a_n(x)y^{(n)} + ... + a_1(x)y' + a_0(x)y = g(x)$. Coefficients are functions of $x$, equation is first-degree in $y$.

11. Which function is a solution to $y' = 3x^2$?

    Any antiderivative of $3x^2$, e.g., $y = x^3 + C$. Integrate $3x^2$ to get $x^3 + C$.

12. Verify if $y = e^{3x}$ solves $y' = 3y$.

    Yes, $y' = 3e^{3x}$. Derivative matches 3 times the function.

13. Determine if $y = \text{cos}(x)$ satisfies $y'' + y = 0$.

    Yes, $y'' + y = 0$. $y'' = -\cos x$, so $y'' + y = 0$.

14. State the general solution for $y' = k$.

    $y = kt + C$. Integrate the constant $k$ with respect to time.

15. Identify the solution form for $y'' + y = 0$.

    General solution: $y = C_1 \text{cos}(x) + C_2 \text{sin}(x)$. Characteristic equation $r^2 + 1 = 0$ gives $r = \pm i$.

16. What is a solution to a differential equation?

    A function satisfying the differential equation. When substituted, makes the equation true.

17. Determine if $y = \text{sin}(x)$ solves $y'' + y = 0$.

    Yes, $y'' + y = 0$. $y'' = -\sin x$, so $y'' + y = 0$.

18. Verify $y = x^3$ for $y'' = 6x$.

    Yes, $y'' = 6x$. Second derivative of $x^3$ is $6x$.

19. Is $y = Ce^{3x}$ a solution for $y' = 3y$?

    Yes, $y' = 3y$. Derivative of $Ce^{3x}$ is $3Ce^{3x} = 3y$.

20. What is the solution form for $y'' + 9y = 0$?

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

21. Verify if $y = e^x$ satisfies $y' = y$.

    Yes, $y' = e^x = y$. Derivative of $e^x$ equals itself.

22. What is the solution for $y' = 0$?

    $y = C$. Derivative of constant is zero.

23. What is the first step in verifying a solution?

    Substitute the function into the differential equation. Direct substitution method for verification.

24. Verify if $y = Ce^{-2x}$ solves $y' + 2y = 0$.

    Yes, it satisfies $y' + 2y = 0$. $y' = -2Ce^{-2x} = -2y$, so $y' + 2y = 0$.

25. Is $y = x^2 + C$ a solution to $y'' = 2$?

    Yes, $y'' = 2$. Second derivative of $x^2 + C$ is constant 2.

26. Verify $y = 2x - 1$ for $y' = 2$.

    Yes, $y' = 2$. Derivative of linear function $2x - 1$ is 2.

27. Verify if $y = x^2$ satisfies $y'' = 2$.

    Yes, $y'' = 2$. Second derivative of $x^2$ is constant 2.

28. Find the derivative of $y = \frac{1}{x}$.

    $y' = -\frac{1}{x^2}$. Power rule: $\frac{d}{dx}[x^{-1}] = -x^{-2}$.

29. What is the next step after substitution in solution verification?

    Check if the equation holds true. Verify both sides are equal after substitution.

30. What is the general solution for $y'' - 4y = 0$?

    $y = C_1 e^{2x} + C_2 e^{-2x}$. Characteristic equation $r^2 - 4 = 0$ gives $r = \pm 2$.

31. Verify if $y = Cx^{-1}$ is a solution for $xy' + y = 0$.

    Yes, it satisfies $xy' + y = 0$. $y' = -Cx^{-2}$, and $xy' + y = 0$ checks out.

32. Verify if $y = e^{-2x}$ satisfies $y' = -2y$.

    Yes, $y' = -2e^{-2x}$. Derivative equals -2 times the function.

33. Determine if $y = \frac{1}{x}$ solves $xy' = -y^2$.

    Yes, $y' = -\frac{1}{x^2}$. Check: $x(-\frac{1}{x^2}) = -\frac{1}{x} = -(\frac{1}{x})^2$.

34. Determine if $y = e^{-x}$ solves $y' = -y$.

    Yes, $y' = -e^{-x}$. Derivative equals negative of the function.

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