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Kernel: SageMath 9.0
x = var('x') y = function('y')(x) ED = diff(y,x,5)+5*diff(y,x,4)-2*diff(y,x,3)-10*diff(y,x,2)+diff(y,x)+5*y == 0 desolve(ED,y,contrib_ode=true,algorithm='fricas').show()
C3xe(x)+C1xex+C2e(x)+C4e(5x)+C0ex\renewcommand{\Bold}[1]{\mathbf{#1}}_{C_{3}} x e^{\left(-x\right)} + _{C_{1}} x e^{x} + _{C_{2}} e^{\left(-x\right)} + _{C_{4}} e^{\left(-5 \, x\right)} + _{C_{0}} e^{x}
x = var('x') y = function('y')(x) y1 = function('y1')(x) y2 = function('y2')(x) y3 = function('y3')(x) y4 = function('y4')(x) eq1 = y1 == diff(y, x) eq2 = y2 == diff(y1, x) eq3 = y3 == diff(y2, x) eq4 = y4 == diff(y3, x) eq5 = diff(y4,x) + 5*y4 - 2*y3 - 10*y2 + y1 + 5*y == 0 sol = desolve_system([eq1, eq2, eq3, eq4, eq5], [y,y1,y2,y3,y4], algorithm='fricas') sol[0]
y(x) == _C3*x*e^(-x) + _C1*x*e^x + _C2*e^(-x) + _C4*e^(-5*x) + _C0*e^x
x = var('x') y = function('y')(x) y1 = function('y1')(x) y2 = function('y2')(x) y3 = function('y3')(x) y4 = function('y4')(x) eq1 = y1 == diff(y, x) eq2 = y2 == diff(y1, x) eq3 = y3 == diff(y2, x) eq4 = y4 == diff(y3, x) eq5 = diff(y4,x)+5*y4-2*y3-10*y2+y1+5*y == 0 sol = desolve_system([eq1, eq2, eq3, eq4, eq5], [y,y1,y2,y3,y4]) sol
[y(x) == 5/16*x*e^(-x)*y(0) - 5/24*x*e^x*y(0) - 1/4*x*e^(-x)*y1(0) - 1/4*x*e^x*y1(0) - 3/8*x*e^(-x)*y2(0) + 1/6*x*e^x*y2(0) + 1/4*x*e^(-x)*y3(0) + 1/4*x*e^x*y3(0) + 1/16*x*e^(-x)*y4(0) + 1/24*x*e^x*y4(0) + 1/64*(35*y(0) - 48*y1(0) - 6*y2(0) + 16*y3(0) + 3*y4(0))*e^(-x) + 1/576*(y(0) - 2*y2(0) + y4(0))*e^(-5*x) + 1/144*(65*y(0) + 108*y1(0) + 14*y2(0) - 36*y3(0) - 7*y4(0))*e^x, y1(x) == -5/16*x*e^(-x)*y(0) - 5/24*x*e^x*y(0) + 1/4*x*e^(-x)*y1(0) - 1/4*x*e^x*y1(0) + 3/8*x*e^(-x)*y2(0) + 1/6*x*e^x*y2(0) - 1/4*x*e^(-x)*y3(0) + 1/4*x*e^x*y3(0) - 1/16*x*e^(-x)*y4(0) + 1/24*x*e^x*y4(0) - 1/64*(15*y(0) - 32*y1(0) + 18*y2(0) - y4(0))*e^(-x) - 5/576*(y(0) - 2*y2(0) + y4(0))*e^(-5*x) + 1/144*(35*y(0) + 72*y1(0) + 38*y2(0) - y4(0))*e^x, y2(x) == 5/16*x*e^(-x)*y(0) - 5/24*x*e^x*y(0) - 1/4*x*e^(-x)*y1(0) - 1/4*x*e^x*y1(0) - 3/8*x*e^(-x)*y2(0) + 1/6*x*e^x*y2(0) + 1/4*x*e^(-x)*y3(0) + 1/4*x*e^x*y3(0) + 1/16*x*e^(-x)*y4(0) + 1/24*x*e^x*y4(0) - 1/64*(5*y(0) + 16*y1(0) - 42*y2(0) + 16*y3(0) + 5*y4(0))*e^(-x) + 25/576*(y(0) - 2*y2(0) + y4(0))*e^(-5*x) + 1/144*(5*y(0) + 36*y1(0) + 62*y2(0) + 36*y3(0) + 5*y4(0))*e^x, y3(x) == -5/16*x*e^(-x)*y(0) - 5/24*x*e^x*y(0) + 1/4*x*e^(-x)*y1(0) - 1/4*x*e^x*y1(0) + 3/8*x*e^(-x)*y2(0) + 1/6*x*e^x*y2(0) - 1/4*x*e^(-x)*y3(0) + 1/4*x*e^x*y3(0) - 1/16*x*e^(-x)*y4(0) + 1/24*x*e^x*y4(0) + 1/64*(25*y(0) - 66*y2(0) + 32*y3(0) + 9*y4(0))*e^(-x) - 125/576*(y(0) - 2*y2(0) + y4(0))*e^(-5*x) - 1/144*(25*y(0) - 86*y2(0) - 72*y3(0) - 11*y4(0))*e^x, y4(x) == 5/16*x*e^(-x)*y(0) - 5/24*x*e^x*y(0) - 1/4*x*e^(-x)*y1(0) - 1/4*x*e^x*y1(0) - 3/8*x*e^(-x)*y2(0) + 1/6*x*e^x*y2(0) + 1/4*x*e^(-x)*y3(0) + 1/4*x*e^x*y3(0) + 1/16*x*e^(-x)*y4(0) + 1/24*x*e^x*y4(0) - 1/64*(45*y(0) - 16*y1(0) - 90*y2(0) + 48*y3(0) + 13*y4(0))*e^(-x) + 625/576*(y(0) - 2*y2(0) + y4(0))*e^(-5*x) - 1/144*(55*y(0) + 36*y1(0) - 110*y2(0) - 108*y3(0) - 17*y4(0))*e^x]