Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 29

Warm-up-exercises

Exercise

Find the solutions to the ordinary differential equation

${}y'=-{\frac {y}{t}}\,.$

Exercise

Find the solutions to the ordinary differential equation

${}y'={\frac {y}{t^{2}}}\,.$

Exercise

Find the solutions to the ordinary differential equation

${}y'=e^{t}y\,.$

Exercise

Find the solutions of the inhomogeneous linear differential equation

${}y'=y+7\,.$

Exercise

Find the solutions to the inhomogeneous linear differential equation

${}y'=y+{\frac {\sinh t}{\cosh ^{2}t}}\,.$

Exercise

Let

$f\colon I\longrightarrow \mathbb {R} _{+}$

be a differentiable function on the interval ${}I\subseteq \mathbb {R}$. Find a homogeneous linear ordinary differential equation for which ${}f$ is a solution.

Exercise

Let

${}y'=g(t)y\,$

be a homogeneous linear ordinary differential equation with a function ${}g$ differentiable infinitely many times and let ${}y$ be a differentiable solution.

a) Prove that ${}y$ is also infinitely differentiable.

b) Let ${}y(t_{0})=0$ for a time-point ${}t_{0}$. Prove, using the formula

${}(f\cdot g)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}f^{(k)}\cdot g^{(n-k)}\,,$

that ${}y^{(n)}(t_{0})=0$ for all ${}n\in \mathbb {N}$.

Exercise

a) Find all solutions for the ordinary differential equation (${}t\in \mathbb {R}$)

${}y'={\frac {y}{t}}\,.$

b) Find all solutions for the ordinary differential equation (${}t\in \mathbb {R}$)

${}y'={\frac {y}{t}}+t^{7}\,.$

c) Solve the initial value problem

$y'={\frac {y}{t}}+t^{7}{\text{ and }}y(1)=5.$

The following statement is called the superposition principle for inhomogeneous linear differential equations. It says in particular that the difference of two solutions of an inhomogeneous linear differential equation is a solution of the corresponding homogeneous linear differential equation.

Exercise

Let ${}I\subseteq \mathbb {R}$ be a real interval and let

$g,h_{1},h_{2}\colon I\longrightarrow \mathbb {R}$

be functions. Let ${}y_{1}$ be a solution to the differential equation ${}y'=g(t)y+h_{1}(t)$ and let ${}y_{2}$ be a solution to the differential equation ${}y'=g(t)y+h_{2}(t)$. Prove that ${}y_{1}+y_{2}$ is a solution to the differential equation

${}y'=g(t)y+h_{1}(t)+h_{2}(t)\,.$

Hand-in-exercises

Exercise

Confirm by computation that the function

${}y(t)=c{\frac {\sqrt {t-1}}{\sqrt {t+1}}}\,$

found in the example satisfies the differential equation

${}y'=y/(t^{2}-1)\,.$

Exercise

Find the solutions to the ordinary differential equation

${}y'={\frac {y}{t^{2}-3}}\,.$

Exercise

Solve the initial value problem

$y'={\frac {t}{t^{2}+2}}y{\text{ with }}y(3)=7$

Exercise

Find the solutions to the inhomogeneous linear differential equation

${}y'=y+e^{2t}-4e^{-3t}+1\,.$

Exercise

Find the solutions to the inhomogeneous linear differential equation

${}y'={\frac {y}{t}}+{\frac {t^{3}-2t+5}{t^{2}-3}}\,.$