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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 about Ordinary and Regular points and Method of Frobenious.
- Current Lecture Note --> Note 10 is about Bessel Equation.
### Quizzes

- Solution to Quiz 1.
- Solution to Practice Exam 1 - Quiz 2.
- Solution to Quiz 3.
### Exams

- Solution to Exam 1.
### Supplementary Problems and Practice Exams

- Supplementary Problems for Exam 1. There is a typo on Question 3. Now it is corrected. Let me know if there are more typos.
- Practice Exam 2 and Old Exam 2(Given in fall 2016)
### 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 18~~Friday, October 20 by the class- 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

- 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 $b_1$.

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