Why Matlab could not solve a set of linear differential equations with initial conditions through dsolve?

Hi,

Where is the problem in my codes to solve a set of linear differential equations with initial conditions?

Any suggest?

clc
clear

ML = [2.53735261480440e-10	-1.35406667270221e-16	1.30871725825994e-18	-2.13374675288863e-15	2.03261768716403e-17	-1.61477754970584e-16	-1.62541250675724e-16;
-1.35406667270221e-16	2.53734628229043e-10	-2.73582082145325e-17	4.72489571641653e-16	1.92706132643316e-16	8.27766626176449e-16	-4.33522241607763e-16;
1.30871725825994e-18	-2.73582082145325e-17	2.53811694377620e-10	-1.38664722187494e-14	1.89471206077083e-13	-5.47669811839268e-14	2.67184290813528e-14;
-2.13374675288863e-15	4.72489571641653e-16	-1.38664722187494e-14	2.53715763347126e-10	-6.05035598297774e-15	-2.32344311157266e-14	2.96103830935012e-14;
2.03261768716403e-17	1.92706132643316e-16	1.89471206077083e-13	-6.05035598297774e-15	2.53850804700165e-10	-6.82222668978547e-14	4.82767908961758e-14;
-1.61477754970584e-16	8.27766626176449e-16	-5.47669811839268e-14	-2.32344311157266e-14	-6.82222668978547e-14	2.53705685445938e-10	3.43519159646703e-14;
-1.62541250675724e-16	-4.33522241607763e-16	2.67184290813528e-14	2.96103830935012e-14	4.82767908961758e-14	3.43519159646703e-14	2.53690028388901e-10];

KL = [6.34368385323866e-05	-5.80205835437760e-08	-1.23764005930491e-10	-1.48354797737571e-07	-3.76339592244736e-08	-1.55516201961077e-07	-2.99714848304693e-07;
-5.80205835437760e-08	0.00277423390772568	-3.64032926965371e-10	-1.15677999366471e-06	-5.05159524190059e-07	-1.14803879509422e-06	-2.36449328010142e-06;
-1.23764005930491e-10	-3.64032926965371e-10	0.00351253253565176	2.60553493225360e-07	-2.50880577858935e-05	8.96289108975872e-06	2.53253964466067e-07;
-1.48354797737571e-07	-1.15677999366471e-06	2.60553493225360e-07	0.0201021660593748	-0.000292315163965329	0.000101788069022335	-6.31134009698694e-06;
-3.76339592244736e-08	-5.05159524190059e-07	-2.50880577858935e-05	-0.000292315163965329	0.0414826409114255	0.00570625287068793	0.00236501515779697;
-1.55516201961077e-07	-1.14803879509422e-06	8.96289108975872e-06	0.000101788069022335	0.00570625287068793	0.0634851951853710	-0.000910774592712826;
-2.99714848304693e-07	-2.36449328010142e-06	2.53253964466067e-07	-6.31134009698694e-06	0.00236501515779697	-0.000910774592712826	0.121679411312940];

F=[0.000289760052925726;
0.000537710491736623;
1.24507643858810e-08;
-0.000328931258625777;
-4.51634984307082e-05;
-0.000110003762488177;
-0.000505551160254736];


syms tau_1(t) tau_2(t) tau_3(t) tau_4(t) tau_5(t) tau_6(t) tau_7(t) 
v = transpose([tau_1 tau_2 tau_3 tau_4 tau_5 tau_6 tau_7]);
odes = diff(diff(v)) == -inv(ML) * KL * v;
C = [v(0) == double(0*inv(ML) * [F]) , diff(v(0)) == double(01*inv(ML) * [F])];
dsolve(odes,C)

Answer 1

[Torsten]

The eigenvalues of a polynomial of degree 14 (=degree of ODEs * number of ODEs) are required to get an analytical solution for your problem. But analytical formulae for roots of polynomials only exist up to degree 4.

[Mehdi]

Any other way to solve my problem?

[Torsten]

Sure. But 7 initial conditions are missing for a numerical solution - you need 14 instead of 7.

[Mehdi]

Sorry, I corrected the codes. Now there are 14 ICs.

[Torsten]

ML = [2.53735261480440e-10	-1.35406667270221e-16	1.30871725825994e-18	-2.13374675288863e-15	2.03261768716403e-17	-1.61477754970584e-16	-1.62541250675724e-16;
-1.35406667270221e-16	2.53734628229043e-10	-2.73582082145325e-17	4.72489571641653e-16	1.92706132643316e-16	8.27766626176449e-16	-4.33522241607763e-16;
1.30871725825994e-18	-2.73582082145325e-17	2.53811694377620e-10	-1.38664722187494e-14	1.89471206077083e-13	-5.47669811839268e-14	2.67184290813528e-14;
-2.13374675288863e-15	4.72489571641653e-16	-1.38664722187494e-14	2.53715763347126e-10	-6.05035598297774e-15	-2.32344311157266e-14	2.96103830935012e-14;
2.03261768716403e-17	1.92706132643316e-16	1.89471206077083e-13	-6.05035598297774e-15	2.53850804700165e-10	-6.82222668978547e-14	4.82767908961758e-14;
-1.61477754970584e-16	8.27766626176449e-16	-5.47669811839268e-14	-2.32344311157266e-14	-6.82222668978547e-14	2.53705685445938e-10	3.43519159646703e-14;
-1.62541250675724e-16	-4.33522241607763e-16	2.67184290813528e-14	2.96103830935012e-14	4.82767908961758e-14	3.43519159646703e-14	2.53690028388901e-10];
KL = [6.34368385323866e-05	-5.80205835437760e-08	-1.23764005930491e-10	-1.48354797737571e-07	-3.76339592244736e-08	-1.55516201961077e-07	-2.99714848304693e-07;
-5.80205835437760e-08	0.00277423390772568	-3.64032926965371e-10	-1.15677999366471e-06	-5.05159524190059e-07	-1.14803879509422e-06	-2.36449328010142e-06;
-1.23764005930491e-10	-3.64032926965371e-10	0.00351253253565176	2.60553493225360e-07	-2.50880577858935e-05	8.96289108975872e-06	2.53253964466067e-07;
-1.48354797737571e-07	-1.15677999366471e-06	2.60553493225360e-07	0.0201021660593748	-0.000292315163965329	0.000101788069022335	-6.31134009698694e-06;
-3.76339592244736e-08	-5.05159524190059e-07	-2.50880577858935e-05	-0.000292315163965329	0.0414826409114255	0.00570625287068793	0.00236501515779697;
-1.55516201961077e-07	-1.14803879509422e-06	8.96289108975872e-06	0.000101788069022335	0.00570625287068793	0.0634851951853710	-0.000910774592712826;
-2.99714848304693e-07	-2.36449328010142e-06	2.53253964466067e-07	-6.31134009698694e-06	0.00236501515779697	-0.000910774592712826	0.121679411312940];
F=[0.000289760052925726;
0.000537710491736623;
1.24507643858810e-08;
-0.000328931258625777;
-4.51634984307082e-05;
-0.000110003762488177;
-0.000505551160254736];
ML_invers = inv(ML);
fun = @(t,v)[v(8:14);-ML_invers * KL * v(1:7)];
v0 = [0*ML_invers * F;1*ML_invers * F];
[T,V] = ode15s(fun,[0 0.015],v0);
plot(T,V(:,1))