University differential equations test questions on Green's function theory

📚 Quick Study Guide

• 🔑 Definition: Green's function $G(x, s)$ for a linear differential operator $L$ satisfies $L[G(x, s)] = \delta(x - s)$, where $\delta$ is the Dirac delta function.
• 💡 Properties:
  • Symmetry: $G(x, s) = G(s, x)$ for self-adjoint operators.
  • Continuity: $G(x, s)$ is continuous except at $x = s$.
  • Jump Discontinuity: The derivative of $G(x, s)$ has a jump discontinuity of $1/p(s)$ at $x = s$, where $p(x)$ is the coefficient of the highest order derivative in the differential operator.
• 📝 Construction: Find two linearly independent solutions $y_1(x)$ and $y_2(x)$ of the homogeneous equation $L[y] = 0$. Then, $$G(x, s) = \begin{cases} A y_1(x) y_2(s), & x < s \\ A y_1(s) y_2(x), & x > s \end{cases}$$ where $A$ is a constant determined by the jump condition.
• ⚙️ Solution: The solution to the non-homogeneous equation $L[y] = f(x)$ with appropriate boundary conditions is given by $$y(x) = \int_a^b G(x, s) f(s) ds$$
• 🎯 Boundary Conditions: Green's function must satisfy the given boundary conditions of the problem.

Practice Quiz

1. Question 1: What is the defining property of Green's function $G(x, s)$ for a linear differential operator $L$?
   1. $L[G(x, s)] = 0$
   2. $L[G(x, s)] = 1$
   3. $L[G(x, s)] = \delta(x - s)$
   4. $L[G(x, s)] = x - s$
2. Question 2: For a self-adjoint operator, what symmetry property does Green's function possess?
   1. $G(x, s) = -G(s, x)$
   2. $G(x, s) = G(s, x)$
   3. $G(x, s) = 1/G(s, x)$
   4. $G(x, s) = G(x, -s)$
3. Question 3: Where is Green's function $G(x, s)$ continuous?
   1. Everywhere
   2. Only at $x = s$
   3. Everywhere except at $x = s$
   4. Nowhere
4. Question 4: What is the jump discontinuity in the derivative of $G(x, s)$ at $x = s$, if $p(x)$ is the coefficient of the highest order derivative?
   1. $p(s)$
   2. $1/p(s)$
   3. $-p(s)$
   4. $-1/p(s)$
5. Question 5: If $y_1(x)$ and $y_2(x)$ are linearly independent solutions of $L[y] = 0$, how is Green's function constructed?
   1. $G(x, s) = y_1(x) y_2(s)$ for all $x$ and $s$
   2. $G(x, s) = y_1(x) + y_2(s)$ for all $x$ and $s$
   3. $G(x, s) = \begin{cases} A y_1(x) y_2(s), & x < s \\ A y_1(s) y_2(x), & x > s \end{cases}$
   4. $G(x, s) = \begin{cases} A y_1(x) y_1(s), & x < s \\ A y_2(s) y_2(x), & x > s \end{cases}$
6. Question 6: How is the solution $y(x)$ to $L[y] = f(x)$ obtained using Green's function?
   1. $y(x) = \int_a^b G(x, s) f(x) ds$
   2. $y(x) = \int_a^b G(x, s) f(s) dx$
   3. $y(x) = \int_a^b G(x, s) f(s) ds$
   4. $y(x) = \int_a^b G(s, s) f(s) ds$
7. Question 7: What conditions must Green's function satisfy?
   1. Only the differential equation
   2. Only the boundary conditions
   3. Both the differential equation and boundary conditions
   4. Neither the differential equation nor boundary conditions