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


Exercise

Find the solutions to the ordinary differential equation


Exercise

Find the solutions to the ordinary differential equation


Exercise

Find the solutions of the inhomogeneous linear differential equation


Exercise

Find the solutions to the inhomogeneous linear differential equation


Exercise

Let

be a differentiable function on the interval . Find a homogeneous linear ordinary differential equation for which is a solution.


Exercise

Let

be a homogeneous linear ordinary differential equation with a function differentiable infinitely many times and let be a differentiable solution.

a) Prove that is also infinitely differentiable.

b) Let for a time-point .

Prove, using the formula
that

for all .


Exercise

a) Find all solutions for the ordinary differential equation ()

b) Find all solutions for the ordinary differential equation ()

c) Solve the initial value problem


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 be a real interval and let

be functions. Let be a solution to the differential equation and let be a solution to the differential equation . Prove that is a solution to the differential equation




Hand-in-exercises

Exercise (2 points)

Confirm by computation that the function

found in Example 29.7 satisfies the differential equation


Exercise (3 points)

Find the solutions to the ordinary differential equation


Exercise (5 points)

Solve the initial value problem


Exercise (3 points)

Find the solutions to the inhomogeneous linear differential equation


Exercise (5 points)

Find the solutions to the inhomogeneous linear differential equation




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