114.sagews

var('a, b, c, d, e, M, t1, t2');

(a, b, c, d, e, M, t1, t2)

eqs=[a*e^3+b*e^2+c*e+d==1/e^3,a*M^3+b*M^2+c*M+d==0.5/M^3,5*a*e^3+4*b*e^2+3*c*e+2*d==0,5*a*M^3+4*b*M^2+3*c*M+2*d==(t1+t2)/(2*M)];eqs

[a*e^3 + b*e^2 + c*e + d == 1/e^3, a*M^3 + b*M^2 + c*M + d == 0.500000000000000/M^3, 5*a*e^3 + 4*b*e^2 + 3*c*e + 2*d == 0, 5*a*M^3 + 4*b*M^2 + 3*c*M + 2*d == (t2 + t1)/(2*M)]

s=solve(eqs, a, b, c, d);s

[[a == (4*M^5 - 8*e*M^4 + e^4*((-t2 - t1)*M^3 + 4*M) + e^5*((t2 + t1)*M^2 - 2))/(-2*e^4*M^7 + 6*e^5*M^6 - 6*e^6*M^5 + 2*e^7*M^4), b == (-8*M^6 + 10*e*M^5 - e^4*((-t2 - t1)*M^4 + 5*M^2) + 10*e^2*M^4 - e^5*((-t2 - t1)*M^3 + 5*M) - e^6*((2*t2 + 2*t1)*M^2 - 4))/(-2*e^4*M^7 + 6*e^5*M^6 - 6*e^6*M^5 + 2*e^7*M^4), c == (4*M^7 + 4*e*M^6 - 20*e^2*M^5 + e^5*((-2*t2 - 2*t1)*M^4 + 10*M^2) + e^6*((t2 + t1)*M^3 - 2*M) + e^7*((t2 + t1)*M^2 - 2))/(-2*e^4*M^7 + 6*e^5*M^6 - 6*e^6*M^5 + 2*e^7*M^4), d == (-6*M^6 + 10*e*M^5 - e^5*((-t2 - t1)*M^3 + 5*M) - e^6*((t2 + t1)*M^2 - 3))/(-2*e^3*M^6 + 6*e^4*M^5 - 6*e^5*M^4 + 2*e^6*M^3)]]