
23.3. Analytical Solutions of Mathematical Problems 1347
Example 23.18. Linear fi rst-order system. Exact solu tion. Considering the linear first-order
uncou pled system w ith two independent variables and two dependent variables,
u
t
+ u
x
= sint cos x, v
t
+ 2v
x
= sinx sint,
and applying Mathematica pred efined functions, w e solve this system of PDEs and verify that the
solution obtained (
sol3
),
u(x,t) = −
1
4
cos(x+t)+F
1
(t −x)+
1
2
xsin(t −x), v(x,t) = F
2
(t −x/2)−
1
2
sin(t −x)−
1
6
sin(t +x),
is an exact solution of this linear system as follows:
sys1={D[u[x,t],t]+D[u[x,t],x]==Sin[t]*Cos[x],D[v[x,t],t]+2*D[v[x,t],x]==Sin[x]*Sin[t]}
sol1=DSolve[sys1,{u,v},{x,t}]//First ...