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



Warm-up-exercises

Find the solutions to the ordinary differential equation

 



Find the solutions to the ordinary differential equation

 



Find the solutions to the ordinary differential equation

 



Find the solutions of the inhomogeneous linear differential equation

 



Find the solutions to the inhomogeneous linear differential equation

 



Let

 

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



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  .



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.

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

Confirm by computation that the function

 

found in the example satisfies the differential equation

 



Find the solutions to the ordinary differential equation

 



Solve the initial value problem

 



Find the solutions to the inhomogeneous linear differential equation

 



Find the solutions to the inhomogeneous linear differential equation