130.sagews

7.1. :
$y'-\frac{y}{x}=0$ .

x = var('x')
y = function('y',x)
de = lambda(p): diff(y,x) -y/x
desolve(de(y(x)),[y,x])

'%c*x'

7.2. :
$y'-\frac{y}{x}-x=0$ .

x = var('x')
y = function('y',x)
de = lambda(p): diff(y,x) -y/x -x
desolve(de(y(x)),[y,x])

'x*(x+%c)'

7.3. :
$y''-1+2x=0$ .

x = var('x')
y = function('y',x)
de = lambda(p): diff(y,x,2)-1+2*x
desolve(de(y(x)),[y,x])

'-(2*x^3-3*x^2)/6+%k2*x+%k1'

7.4.
$y''-5y'+6y=0$ .

x = var('x')
y = function('y',x)
de = lambda(p): diff(y,x,2)-5*diff(y,x)+6*y
desolve(de(y(x)),[y,x])

'%k1*%e^(3*x)+%k2*%e^(2*x)'

7.5. :
$y'+y=1$, $y(0)=2$ .

x = var('x')
y = function('y',x)
de = lambda(p): diff(y,x)+y-1
sol=desolve_laplace(de(y(x)),["x","y"],[0,2]);sol

'%e^-x+1'

plot(exp(-x)+1,0,5)

7.6. :
$y''+y=0$, $y(0)=0$, $y'(0)=1$.

x = var('x')
y = function('y', x)
de=lambda(p): diff(y,x,x)+y
desolve_laplace(de(y(x)),["x","y"],[0,0,1])

'sin(x)'

7.7.
$\left\{ \begin{array}{ll} x'_t+y=1\\ y'_t-x=-1\text{,} \end{array} \right.$ $x(0)=1$, $y(0)=-1$.

sage: t = var('t')
sage: x = function('x', t)
sage: y = function('y', t)
sage: de1 = lambda z: diff(z[0],t) + z[1] - 1
sage: de2 = lambda z: diff(z[1],t) - z[0] + 1
sage: des = [de1([x(t),y(t)]),de2([x(t),y(t)])]
sage: vars = ["t","x","y"]
sage: desolve_system(des,vars)

['sin(t)+cos(t)+1', 'sin(t)-cos(t)+1']

desolve_system(des,vars,[0,1,-1])

['2*sin(t)+1', '1-2*cos(t)']

7.8. $y'=x^2+y^2$.

maxima.eval('load("plotdf")')
maxima.eval('plotdf(x^2+y^2,[x,-3,3],[y,-3,3])')

'2'

7.9.
$y'=x^2+y^2$, $\;y(0)=1$.

maxima.eval('load("plotdf")')
maxima.eval('plotdf(x^2+y^2,[trajectory_at,0,1],[x,-3,3],[y,-3,3])')

'2'