# Differential Equations

Below you will find the exercises we suggest you work on in connection with the Differential Equations theme and the project problems which are to be done for the exercise classes during November **11-15**.

## Suggested Exercises

**Section 7.9: First-Order Differential Equations**

7.9.1, 7.9.6, **7.9.7**, **7.9.11**, 7.9.13, **7.9.19**, **7.9.21**, 7.9.28, 7.9.31.

**Section 18.1: Classifying Differential Equations**

**18.1.1**, 18.1.3.

**Section 18.3: Existence, Uniqueness, and Numerical Methods**

18.3.1-3, **18.3.4-6**, **18.3.13**, 18.3.15, 18.3.16.

## Project Problems

### Week 46

These problems are to be presented during November **11-15**. See here for where you should meet, and here to sign up for presenting a problem.

**Problem 1: Maple TA**

Given the initial value problem

\[y' +\frac{y}{\tanh{x}} = \ln{2}\cdot\cosh(x) , \ y(1)=b, \]

find the value of b that ensures that y(0) exists.

HINT: Integrating factor, Method 1 page 450 in Adams, and -:Ordinary differential equations.

**Problem 2: Exam 1997 in SIF 5003, problem 5**

The freezing point \(T\) for water with ionic concentration \(x\), satisfies the differential equation

\[\frac{dT}{dx} =\frac{-aT^2}{1+bx}, \ T(0)=T_0, \]

where \(a\), \(b\) and \(T_0\) are constants with values \(a = 2.49\cdot 10^{-5} \, \mathrm{K}^{-1} \mathrm{M}^{-1}\), \(b= 0.018 \, \mathrm{M}^{-1}\) and \(T_0 = 273.15 \, \mathrm{K}\), where \(\mathrm{M}=\) mol and \(\mathrm{K}=\) Kelvin.

a) Use the differential equation to find the tangent to \(T(x)\) in the point \((0, T_0)\). Use this tangent to find an approximate value to \(T(1.2)\).

b) Solve the initial value problem. Use the solution to find \(T(1.2)\), and compare this value with the value you found in a).

HINT 1: -:Ordinary differential equations.

HINT 2: Differential equations videos.

**Problem 3: Continuation Exam 2006 in TMA4100, problem 5**

Consider the initial value problem

\[\frac{dy}{dx} = x+y^2,\] with \(y(0)=1\).

a) Use Euler's Method with step-length \(h=0.1\) to find an approximation to \(y(0.3).\)

HINT 1: :Numerical methods.

HINT 2: Differential equations videos.

b) Let \(P_2(x)\) denote the second order Taylor polynomial for the solution of the initial value problem \(y(x)\) at \(x=0\). Find \(P_2(0.3)\). HINT: Differentiate the differential equation implicitly to find \(y''\).

**Problem 4: Modeling fish population**

A lake has a population of \(x(t)\) fish at time \(t\). Assume that the probability of one fish meeting another in a small time interval is proportional with the population size.

a) Assume that the rate of birth in the population is proportional with the number of random encounters between two fish, and that the rate of death is proportional with the population size. Show that these assumptions leads to the equation

\[\frac{dx}{dt}=bx^2-ax,\] where \(a, b\) are positive constants. Give an interpretation on the terms \(bx^2\) and \(ax\).

b) Assume that the initial population is \(x(0)=x_0\), and find \(x(t)\).

c) Show that there exists a constant \(k_0\) so that if \(x_0 <k_0\), then the population will eventually die out, while if \(x_0>k_0\) the population will grow infinitely large in finite time. Find \(k_0\) and this finite time. What happens if \(k_0=x_0\)?