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# Differential Equations for Applications

Math 3410, Sections 001 and 002 - Elementary Differential Equations, Fall 2017 [Course Syllabus]

Lectures: MWF 09:05 - 09:55 at LH 201.
Office hours: Monday 12:00 - 13:30, Wednesday 12:30 - 14:00 at MONT 304.
• ### Lecture Notes

• Note 1 -- Note 2 -- Note 3 -- Note 4 -- Note 5(We skipped the applications and we will come back to that later).
• Note 6 is about Autonomous Equations and Stability of Equilibrium Solutions.
• Note 7 is about Secon Order Linear equations and Laplace Transform.
• Note 8 is about Power series solutions of DEs.
• Note 9 is about Ordinary and Regular points and Method of Frobenious.
• Note 10 is about Bessel Equation.
• Note 11 is about Boundary Value Problems and Fourier Series.
• Note 12 is about Partial Differential Equations.
• Note 13 is about Heat Equation.
• Note 14 is about Wave Equation.
• Current Lecture Note --> Note 15 is about Laplace Equation.

• ### Homework

• #### HW1 - Due on Friday, September 8 by the class | Solution to selected problems (PDF)

• Problem 1: Show that $e^{2x}+e^{2y}=1$ is an implicit solution to the DE $e^{x-y}+e^{y-x} \frac{dy}{dx}=0$.
• Problem 2: Find a 1-parameter family of solutions of the DE $y'=y$ and the particular solution for which $y(3)=1$.
• Problem 3: Construct a direction field for the differential equation $y'=2x$.
• Problem 4: Find a particular solution to the DE $y'=e^{x+y}$ with the initial value $y(0)=0$.
• #### HW2 - Due on Friday, September 15 by the class | Solution to selected problems (PDF)

• Problem 1: Check that if the differential equation is exact $(e^{x}\sin y+e^{-y})dx - (xe^{-y} - e^{x} \cos y) dy= 0$. If it is exact then solve the differential equation.
• Problem 2: Check that if the differential equation is exact $e^{x}(x+1) dx +(ye^y - xe^x)dy= 0$. If it is exact then solve the differential equation.
• Problem 3: Let $P(x)=\int p(x) dx$. Show that $e^{P(X)}$ is an integrating factor for the DE
$y'+p(x) y=q(x).$
• Problem 4: Suppose that $a,b,c,e$ are constants such that $ae-bc\neq 0$. Let $m$ and $n$ be arbitrary real numbers. Show that $(ax^{m}y+by^{n+1})dx + (cx^{m+1}+exy^{n})dy=0$ has an integrating factor $\mu(x,y)=x^{\alpha}y^{\beta}$ for some $\alpha$ and $\beta$.
• #### HW3 - Due on Friday, September 22 Monday, September 25 by the class | Solution to selected problems (PDF)

• Problem 1: Find the orthogonal trajectories of the family of circles centered on $x$-axis and passing through the origin.
• Problem 2: Find the orthogonal trajectories of the family of curves having equation $e^x \cos(y)=k$.
• Problem 3: Find the general solution of the Bernoulli equation $xy'+y+x^2y^2e^x=0$.(You may need to rewrite the equation!).
• Problem 4: Find the general solution of the Bernoulli equation $x^2y'+2y=2e^{\frac{1}{x}}y^{\frac{1}{2}}$.
• Problem 5: Find the general solution of the Ricatti equation $y'=1+\frac{y}{x}-\frac{y^2}{x^2}$ with given particular solution $y_1(x)=x$.
• Problem 6: Find the general solution of the Ricatti equation $y'=y^2+2xy+(x^2-1)$ with given particular solution $y_1(x)=-x$.
• #### HW4 - Due on Wednesday, October 4 by the class Problems are from the Supplementary Problems for Exam 1. Here is Solution to selected problems (PDF)

• 1. in Question 23.
• 2., 3. 4. in Question 24.
• 3. in Question 25.
• Question 26.
• Question 28.
• #### HW5 - Due on Wednesday, October 18Friday, October 20 by the class | Solution to selected problems (PDF)

• Find a power series solution $y(x)$ around the point $x_0=0$ to the differential equation $y''+y=0.$ Verify that the power series solution you found has the form $y(x)=a_0\cos(x)+a_1\sin(x).$
• By using the second method find at least first four terms of the power series solution $y(x)$ around the point $x_0=0$ to the differential equation $y''=xy-(y')^2$ with $y(0)=2$ and $y'(0)=1$ (assume that the solution is analytic around $x_0=0$).
• Consider the Rayleigh's equation $my''+ky = ay'-b(y')^{3}$ which models the oscillation of a clarinet reed. Using the second method find the first four terms of the power series solution $y(x)$ around $x_0=0$ with $m=k=a=1$ and $b=1/3$ with the initial conditions $y(0)=0$ and $y'(0)=1$. Write the first four terms of the solution $y(x)$.
• Consider the following differential equation $y''+4 (y^{2}+1)y'+xy=0.$ Use the second method to find first four terms of the power series solution $y(x)=\sum\limits_{n=0}^{\infty} a_n x^n$ around $x_0=0$ with the initial conditions $y(0)=0$ and $y'(0)=1$.
• #### HW6 - Due on Friday, October 27 by the class | Solution to selected problems (PDF)

• For the following differential equation $4xy''+2y'+y=0$
• Find and classify all points as ordinary, regular singular, or irregular singular points.
• For each of the regular point(s), find the corresponding indicial equation and find roots $r_1$ and $r_2$ of the indicial equation (Yes, there are two roots and the difference is not integer).
• Find the corresponding recurrence relations for each of the roots $r_1, r_2$.
• Find the corresponding power series solutions $y_1$ and $y_2$.

• For the following differential equation $xy''+y'-y=0$
• Find and classify all points as ordinary, regular singular, or irregular singular points.
• For each of the regular point(s), find the corresponding indicial equation and find the double root $r_1$of the indicial equation (Yes there is one double root).
• Find the corresponding recurrence relation for the root $r_1$.
• Find the corresponding power series solution $y_1$.
• Use the method of Frobenious and write down the general form of the second solution $y_2$.
• Find at least first two terms of the second solution $b_0$ and $b_1$.

• For the following differential equation $xy''+y=0$
• Find and classify all points as ordinary, regular singular, or irregular singular points.
• For each of the regular point(s), find the corresponding indicial equation and find the roots $r_1$ and $r_2$ of the indicial equation (Yes there are two roots with $r_1-r_2$ is integer).
• Find the corresponding recurrence relation for the roots $r_1$ and $r_2$.
• Find the corresponding power series solution for $y_1$.
• Use the method of Frobenious and write down the general form of the second solution $y_2$.
• Find at least first two terms of the second solution $b_0$ and $b_1$.
• #### HW7 - Due on Monday, November 13 by the class | Solution to selected problems (PDF)

• Let $f(x)$ be given as $f(x)=\left\{ \begin{array}{ll} 0 &-\pi< x < 0, \\ x & 0< x< \pi. \end{array} \right. \quad f(x)=f(x+2\pi).$
• Find the Fourier series $F(x)$ of $f(x)$.
• Using the first part verify that $\frac{\pi}{4}=\sum\limits_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)}.$
• Using the first part verify also that $\frac{\pi^2}{8}=\sum\limits_{n=0}^{\infty} \frac{1}{(2n+1)^{2}}.$

• Let $f(x)$ be given as $f(x)=\left\{ \begin{array}{ll} x &0\leq x\leq \frac{1}{2}\pi, \\ \pi-x & \frac{1}{2}\pi\leq x\leq \pi. \end{array} \right.$
• Extend $f(x)$ into an odd periodic function with period of $2\pi$ and find its Fourier series $F(x)$.
• Extend $f(x)$ into an even periodic function with period of $2\pi$ and find its Fourier series $F(x)$.
• Using either the first part or the second part verify that $\frac{\pi^{2}}{8}=\sum\limits_{n=1}^{\infty} \frac{1}{(2n-1)^{2}}.$
• #### HW8 - Due on Wednesday, November 29 by the class Solution to selected problems (PDF)

• Problem 1: Consider the following Heat conduction problem $\left\{ \begin{array}{l} u_{xx}=u_{t}, \quad 0 < x < 2, \quad t > 0,\\ u(0,t)=0 \quad \mbox{and} \quad u(2,t)=0,\\ u(x,0)=3\sin(\pi x)-4\sin(\frac{3\pi x}{2}). \end{array} \right.$
• By considering separation of variables $u(x,t)=X(x)T(t)$, rewrite the partial differential equation
in terms of two ordinary differential equations in $X$ and $T$ (take arbitrary constant as $-\lambda$).
• Rewrite the boundary values in terms of $X$ and $T$.
• Now choose the boundary values which will not give a non-trivial solution and write the ordinary differential equation corresponding to $X$.
• Solve the two-point boundary value problem corresponding to $X$. Find all eigenvalues $\lambda_n$ and eigenfunctions $X_n$.
• For each eigenvalue $\lambda_n$ you found, rewrite and solve the ordinary differential equation corresponding to $T_n$.
• Now write general solution for each $n$, $u_n(x,t)=X_n(x) T_n(t)$ and find the general solution $u(x,t)=\sum u_n(x,t)$.
• Using the given initial value and the general solution you found, find the particular solution.

• Problem 2: Consider the following Heat conduction problem $\left\{ \begin{array}{l} 9u_{xx}=u_{t}, \quad 0 < x < 3, \quad t > 0,\\ u_{x}(0,t)=0 \quad \mbox{and} \quad u_{x}(3,t)=0,\\ u(x,0)=2\cos(\frac{\pi x}{3}) -4 \cos(\frac{5\pi x}{3}). \end{array} \right.$
• By considering separation of variables $u(x,t)=X(x)T(t)$, rewrite the partial differential equation
in terms of two ordinary differential equations in $X$ and $T$ (take arbitrary constant as $-\lambda$).
• Rewrite the boundary values in terms of $X$ and $T$.
• Now choose the boundary values which will not give a non-trivial solution and write the ordinary differential equation corresponding to $X$.
• Solve the two-point boundary value problem corresponding to $X$. Find all eigenvalues $\lambda_n$ and eigenfunctions $X_n$.
• For each eigenvalue $\lambda_n$ you found, rewrite and solve the ordinary differential equation corresponding to $T_n$.
• Now write general solution for each $n$, $u_n(x,t)=X_n(x) T_n(t)$ and find the general solution $u(x,t)=\sum u_n(x,t)$.
• Using the given initial value and the general solution you found in, find the particular solution.
• #### HW9 - Due on Wednesday, December 6 Friday, December 8 by the class Solution to selected problems (PDF)

• Problem: Consider the following Wave equation which describes the displacement $u(x, t)$ of a piece of flexible string with the initial boundary value problem $\left\{ \begin{array}{l} 25u_{xx}=u_{tt}, \quad 0 < x < 5, \quad t > 0,\\ u(0,t)=0 \quad \mbox{and} \quad u(5,t)=0,\\ u(x,0)=0 \quad \mbox{and} \quad u_{t}(x,0)=3\sin(\frac{3\pi x}{5})-10\sin(\frac{4\pi x}{5}). \end{array} \right.$
• By considering separation of variables $u(x,t)=X(x)T(t)$, rewrite the partial differential equation
in terms of two ordinary differential equations in $X$ and $T$.
• Rewrite the boundary values in terms of $X$ and $T$.
• Now choose the boundary values which will not give a non-trivial solution and then rewrite the ordinary differential equation corresponding to $X$.
• Solve the two-point boundary value problem corresponding to $X$ you found. Find all eigenvalues $\lambda_n$ and eigen functions $X_n$
• For each eaigenvalue $\lambda_n$ you found, solve the initial value problem corresponding to $T_n$.
• Now write general solution for each $n$, $u_n(x,t)=X_n(x) T_n(t)$ and find the general solution $u(x,t)=\sum u_n(x,t)$.
• Using the given initial values, find the particular solution.