# NDSolve ERROR: There are more dependent variables, than equations, so the system is underdetermined

I have a question concerning NDSolve and a system of differential equations. I have 4 variables and 4 equations. Why do I get this ERROR :There are more dependent variables, {f[j,p],U[j,p],(f^(0,1))[j,p],(U^(0,1))[j,p]}, than equations, so the system is underdetermined. ???

V = 1000; ρ = 0.000013; L = 37;

eqns = {V p D[f[j, p], j] +  j D[U[j, p], j] +  U[j, p] + 6  ρ L j == 0,
V f[j, p] + V p D[f[j, p], p] + j D[U[j, p], p] == 0};

vars = {f[j, p], U[j, p]};

inits = {f[5, 10] == 0.1048, U[5, 10] == 3.6};

res = NDSolve[{eqns, inits}, vars, {j, p}]

• A little tough reading your work, however I believe your problem may arise from using "=="; this tests if the left and right hand side are equal, it is not used to assign values Commented Apr 23, 2019 at 19:45
• This set of equations can be solved manually to yield, 0.002886 j^2/2 + j U[j, p] + 1000 p f[j, p] == c, where c is a constant. So, there are too few dependent variables. Commented Apr 23, 2019 at 21:13
• Your initial conditions doesn't make any sense. What are the four variables and four equations?
– zhk
Commented Apr 24, 2019 at 1:31
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To elaborate on my earlier comment, eqns can be rewritten as

eqns1 = {V D[p f[j, p], j] + D[j U[j, p], j] + 6 ρ L j == 0,
V D[p f[j, p], p] + D[j U[j, p], p] == 0};


That the two are equivalent can be verified by

Simplify[eqns == eqns1]
(* True *)


Introducing the new dependent variable,

g[j, p] == V p f[j, p] + j U[j, p]


further simplifies the equations to

eqns2 = {D[g[j, p], j] + 6 ρ L j == 0, D[g [j, p], p] == 0}


Neither DSolve nor NDSolve can handle these equations without human assistance, because there are two equations but only one dependent variable. However, the solution of these two equation clearly is

(* g[j, p] + 3 ρ L j^2 == c *)


with c a constant to be determined by a boundary condition. Inserting inits from the question yields after a small amount of algebra,

{* c -> 1066.036075 *}


Note that g is independent of p`.