Differential Equations Higher-Order Linear Differential Equations Syllabus

Higher-Order Linear Differential Equations syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Higher-Order Linear Differential Equations in Differential Equations, not a summary of it.

1. 1

   Theory of Linear Equations

   4 topics

   • Linear Operators and Superposition
   • Linear Independence and the Wronskian
   • Fundamental Set of Solutions
   • Existence and Uniqueness for Higher-Order Equations
2. 2

   Homogeneous Equations with Constant Coefficients

   2 topics

   • Characteristic Equation
   • Reduction of Order
3. 3

   Nonhomogeneous Equations

   2 topics

   • Method of Undetermined Coefficients
   • Variation of Parameters
4. 4

   Cauchy-Euler and Special Equations

   2 topics

   • Cauchy-Euler Equation
   • Equations Reducible to Constant Coefficients
5. 5

   Applications: Mechanical and Electrical Vibrations

   4 topics

   • Free Undamped Motion
   • Damped Motion
   • Forced Motion and Resonance
   • LRC Series Circuits

Higher-Order Linear Differential Equations flashcards for Differential Equations

21 of 55 cards from the Higher-Order Linear Differential Equations deck — real questions with worked answers.

1. What is a linear differential operator $L$ of order $n$, and how is a linear ODE written using it?

   $L = a_{n}(x)D^{n} + a_{n-1}(x)D^{n-1} + \cdots + a_{1}(x)D + a_{0}(x)$, where $D = \frac{d}{dx}$. A linear ODE is written compactly as $L(y) = g(x)$.

2. State the two properties that make the operator $L$ linear.

   Additivity and homogeneity: $L(y_{1} + y_{2}) = L(y_{1}) + L(y_{2})$ and $L(\alpha y) = \alpha L(y)$ for any constant $\alpha$. Combined: $L(\alpha y_{1} + \beta y_{2}) = \alpha L(y_{1}) + \beta L(y_{2})$.

3. State the Superposition Principle for a homogeneous linear ODE $L(y)=0$.

   If $y_{1}, y_{2}, \ldots, y_{k}$ are solutions of $L(y)=0$, then any linear combination $c_{1}y_{1} + c_{2}y_{2} + \cdots + c_{k}y_{k}$ is also a solution.

4. For the nonhomogeneous equation $L(y)=g(x)$, how is the general solution structured?

   $y = y_{c} + y_{p}$, where $y_{c}$ (complementary function) is the general solution of $L(y)=0$ and $y_{p}$ is any particular solution of $L(y)=g(x)$.

5. Define the Wronskian of $n$ functions $f_{1}, f_{2}, \ldots, f_{n}$.

   $$W(f_{1},\ldots,f_{n}) = \begin{vmatrix} f_{1} & f_{2} & \cdots & f_{n} \\ f_{1}' & f_{2}' & \cdots & f_{n}' \\ \vdots & \vdots & & \vdots \\ f_{1}^{(n-1)} & f_{2}^{(n-1)} & \cdots & f_{n}^{(n-1)} \end{vmatrix}$$

6. How does the Wronskian test for linear independence of solutions of a linear ODE on an interval $I$?

   If $y_{1},\ldots,y_{n}$ are solutions of an $n$th-order linear homogeneous ODE on $I$, they are linearly independent on $I$ if and only if $W(y_{1},\ldots,y_{n}) \neq 0$ for every $x$ in $I$.

7. For solutions of a linear ODE, what does Abel's theorem say about where the Wronskian can vanish?

   The Wronskian of solutions is either identically zero on $I$ or never zero on $I$; it cannot vanish at some points and not others.

8. State Abel's formula for the Wronskian of two solutions of $y'' + P(x)y' + Q(x)y = 0$.

   $W(x) = C\, e^{-\int P(x)\,dx}$, where $C$ is a constant. If $C \neq 0$ the solutions are independent.

9. Define a fundamental set of solutions for an $n$th-order homogeneous linear ODE.

   A set of $n$ solutions $y_{1}, y_{2}, \ldots, y_{n}$ that are linearly independent on the interval $I$.

10. Given a fundamental set $\{y_{1},\ldots,y_{n}\}$, what is the general solution of $L(y)=0$?

    $y = c_{1}y_{1} + c_{2}y_{2} + \cdots + c_{n}y_{n}$, where $c_{1},\ldots,c_{n}$ are arbitrary constants.

11. Does a fundamental set of solutions always exist for an $n$th-order homogeneous linear ODE with continuous coefficients?

    Yes. On any interval $I$ where the coefficients (and leading coefficient nonzero) are continuous, a fundamental set of $n$ linearly independent solutions always exists.

12. State the Existence and Uniqueness theorem for the IVP of an $n$th-order linear ODE.

    If $a_{n}(x),\ldots,a_{0}(x)$ and $g(x)$ are continuous on $I$ and $a_{n}(x)\neq0$ on $I$, then for any $x_{0}\in I$ and prescribed values $y(x_{0}), y'(x_{0}), \ldots, y^{(n-1)}(x_{0})$, a unique solution exists on $I$.

13. How many initial conditions are needed to specify a unique solution of an $n$th-order linear ODE, and what are they?

    Exactly $n$ conditions at a single point $x_{0}$: the values $y(x_{0}), y'(x_{0}), \ldots, y^{(n-1)}(x_{0})$.

14. For a homogeneous linear IVP $L(y)=0$ with all $n$ initial conditions equal to zero, what is the solution?

    The trivial solution $y \equiv 0$, and by uniqueness it is the only solution.

15. How is the characteristic (auxiliary) equation obtained from $a_{n}y^{(n)}+\cdots+a_{1}y'+a_{0}y=0$ with constant coefficients?

    Substitute $y=e^{mx}$ to get $a_{n}m^{n} + a_{n-1}m^{n-1} + \cdots + a_{1}m + a_{0} = 0$.

16. For a constant-coefficient ODE, what solution corresponds to a real root $m$ of multiplicity 1 of the characteristic equation?

    The single solution $y = e^{mx}$.

17. For a real root $m$ of the characteristic equation with multiplicity $k$, what are the corresponding linearly independent solutions?

    $e^{mx}, \; x e^{mx}, \; x^{2} e^{mx}, \; \ldots, \; x^{k-1} e^{mx}$.

18. For a simple pair of complex conjugate roots $m = \alpha \pm \beta i$, what are the two real solutions?

    $y_{1} = e^{\alpha x}\cos(\beta x)$ and $y_{2} = e^{\alpha x}\sin(\beta x)$.

19. For complex conjugate roots $\alpha \pm \beta i$ of multiplicity $k$, list the corresponding real solutions.

    $e^{\alpha x}\cos\beta x, \; e^{\alpha x}\sin\beta x, \; x e^{\alpha x}\cos\beta x, \; x e^{\alpha x}\sin\beta x, \; \ldots, \; x^{k-1}e^{\alpha x}\cos\beta x, \; x^{k-1}e^{\alpha x}\sin\beta x$.

20. Solve $y'' - 5y' + 6y = 0$ (state the general solution).

    Characteristic equation $m^{2}-5m+6=(m-2)(m-3)=0$, so $m=2,3$. General solution: $y = c_{1}e^{2x} + c_{2}e^{3x}$.

21. Solve $y'' + 4y = 0$ (state the general solution).

    Roots $m = \pm 2i$ ($\alpha=0,\beta=2$). General solution: $y = c_{1}\cos 2x + c_{2}\sin 2x$.

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