# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 29/en

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Warm-up-exercises

### Exercise

Find the solutions to the ordinary differential equation

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

### Exercise

Find the solutions to the ordinary differential equation

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

### Exercise

Find the solutions to the ordinary differential equation

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

### Exercise

Find the solutions of the inhomogeneous linear differential equation

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

### Exercise

Find the solutions to the inhomogeneous linear differential equation

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

### Exercise

Let

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

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

### Exercise

Let

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

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

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

b) Let ${\displaystyle {}y(t_{0})=0}$ for a time-point ${\displaystyle {}t_{0}}$.

Prove, using the formula
${\displaystyle (f\cdot g)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}f^{(k)}\cdot g^{(n-k)},}$
that

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

### Exercise

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

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

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

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

c) Solve the initial value problem

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

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

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

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

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

Hand-in-exercises

### Exercise (2 points)

Confirm by computation that the function

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

found in Example 29.7 satisfies the differential equation

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

### Exercise (3 points)

Find the solutions to the ordinary differential equation

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

### Exercise (5 points)

Solve the initial value problem

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

### Exercise (3 points)

Find the solutions to the inhomogeneous linear differential equation

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

### Exercise (5 points)

Find the solutions to the inhomogeneous linear differential equation

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