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

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\setcounter{section}{29}






\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Find the solutions to the ordinary differential equation
\mathdisp {y' = - \frac{y}{t}} { . }

}
{} {}




\inputexercise
{}
{

Find the solutions to the ordinary differential equation
\mathdisp {y' = { \frac{ y }{ t^2 } }} { . }

}
{} {}




\inputexercise
{}
{

Find the solutions to the ordinary differential equation
\mathdisp {y' = e^t y} { . }

}
{} {}




\inputexercise
{}
{

Find the solutions of the inhomogeneous linear differential equation
\mathdisp {y' = y + 7} { . }

}
{} {}




\inputexercise
{}
{

Find the solutions to the inhomogeneous linear differential equation
\mathdisp {y' = y + { \frac{ \sinh t }{ \cosh^{ 2 } t } }} { . }

}
{} {}




\inputexercise
{}
{

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

}
{} {}




\inputexercise
{}
{

Let
\mathdisp {y'=g(t) y} { }
be a homogeneous linear ordinary differential equation with a function
\mathl{g}{} differentiable infinitely many times and let $y$ be a differentiable solution.

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

b) Let
\mathl{y(t_0)=0}{} for a time-point $t_0$. Prove, using the formula
\mathdisp {(f \cdot g)^{(n)}= \sum_{k=0}^n \binom { n } { k} f^{(k)} \cdot g^{(n-k)}} { , }
that
\mathl{y^{(n)}(t_0)=0}{} for all
\mathl{n \in \N}{.}

}
{} {}




\inputexercise
{}
{

a) Find all \definitionsverweis {solutions}{}{} for the \definitionsverweis {ordinary differential equation}{}{} \zusatzklammer {\mathlk{t \in \R_+}{}} {} {}
\mathdisp {y' = { \frac{ y }{ t } }} { . }

b) Find all \definitionsverweis {solutions}{}{} for the \definitionsverweis {ordinary differential equation}{}{} \zusatzklammer {\mathlk{t \in \R_+}{}} {} {}
\mathdisp {y' = { \frac{ y }{ t } } +t^7} { . }

c) Solve the initial value problem
\mathdisp {y' = { \frac{ y }{ t } } +t^7 \text{ and } y(1)= 5} { . }

}
{} {}

The following statement is called the \stichwort {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.


\inputexercise
{}
{

Let
\mathl{I \subseteq \R}{} be a real interval and let \maabbdisp {g,h_1,h_2} {I} {\R } {} be functions. Let $y_1$ be a solution to the differential equation
\mathl{y'= g(t) y +h_1(t)}{} and let $y_2$ be a solution to the differential equation
\mathl{y'= g(t) y +h_2(t)}{.} Prove that
\mathl{y_1+y_2}{} is a solution to the differential equation
\mathdisp {y'= g(t)y +h_1(t) +h_2(t)} { . }

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{2}
{

Confirm by computation that the function
\mathdisp {y(t)= c \frac{\sqrt{t-1} }{\sqrt{t+1} }} { }
found in Example 29.7 satisfies the differential equation
\mathdisp {y'=y/(t^2-1)} { . }

}
{} {}




\inputexercise
{3}
{

Find the solutions to the ordinary differential equation
\mathdisp {y' = \frac{y}{t^2-3}} { . }

}
{} {}




\inputexercise
{5}
{

Solve the initial value problem
\mathdisp {y'= \frac{t}{t^2+2} y \text{ with } y(3) = 7} { . }

}
{} {}




\inputexercise
{3}
{

Find the solutions to the inhomogeneous linear differential equation
\mathdisp {y' = y+e^{2t}-4e^{-3t}+1} { . }

}
{} {}




\inputexercise
{5}
{

Find the solutions to the inhomogeneous linear differential equation
\mathdisp {y' = { \frac{ y }{ t } } + { \frac{ t^3-2t+5 }{ t^2-3 } }} { . }

}
{} {}



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