DGL_Loesen.sagews

# Uebung 7, Aufgabe 4 c
x = var('x'); y = function('y')
DE = diff(y(x), x) == y(x)*x/(1+x^2)

s=desolve(DE, [y(x),x],show_method=True)
show(s)

[$\displaystyle C e^{\left(\frac{1}{2} \, \log\left(x^{2} + 1\right)\right)}$, linear]

# Uebung 7, Aufgabe 4 e
x = var('x'); y = function('y')
DE = diff(y(x), x)*sin(y(x)) == -x

s=desolve(DE, [y(x),x],contrib_ode=True,show_method=True)
show(s)

[$\displaystyle \cos\left(y\left(x\right)\right) = \frac{1}{2} \, x^{2} + C$, separable]

# Uebung 8, Aufgabe 1 e
x = var('x'); y = function('y')
DE = diff(y(x), x) == 1-y(x)/x
s=desolve(DE, [y(x),x],contrib_ode=True,show_method=True)
show(s)

[$\displaystyle \frac{x^{2} + 2 \, C}{2 \, x}$, linear]

x = var('x')
f(x)=x/(x-1)
F=integrate(f,x)
show(F)

$\displaystyle x \ {\mapsto}\ x + \log\left(x - 1\right)$

# Uebung 8, Aufgabe 2b
x = var('x'); y = function('y')
DE = diff(y(x), x) == x-y(x)
s=desolve(DE, [y(x),x],contrib_ode=True,show_method=True)
show(s)

[$\displaystyle {\left({\left(x - 1\right)} e^{x} + C\right)} e^{\left(-x\right)}$, linear]

# Uebung 8, Aufgabe 1d
x = var('x'); y = function('y')
DE = diff(y(x), x) == 1-y(x)/x
s=desolve(DE, [y(x),x],contrib_ode=True,show_method=True)
show(s)

[$\displaystyle \frac{x^{2} + 2 \, C}{2 \, x}$, linear]

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