# How to linearize this equation in two different variables?

How to linearize this equation in two different variables p[t] and q[t,y]:

ein11= -((1/(p[t] + q[t, y])^3)*(4*p[t]*Derivative[0, 1][a][t, y]^2 +   4*q[t, y]*Derivative[0, 1][a][t, y]^2 - 17*a[t, y]*Derivative[0, 1][a][t, y]*
Derivative[0, 1][q][t, y] + 18*a[t, y]*p[t]*Derivative[0, 2][a][t, y] +
18*a[t, y]*q[t, y]*  Derivative[0, 2][a][t, y] + (p[t] + q[t, y])^2*
(Derivative[2][p][t] + (Derivative[1, 0][a][t,
y]*(Derivative[1][p][t] + Derivative[1, 0][q][t, y]))/
a[t, y] + Derivative[2, 0][q][t, y])[
Derivative[1][p][t] + Derivative[1, 0][q][t, y], t]))

I tried:

Normal[Series[ ein11 /. {q -> Function[{t, y}, k*q[t, y]], p -> Function[{t}, ep*p[t]]}, {k, 0, 1}, {ep, 0, 1}]]

But this linearize only one parameter q[t,y] and let unlinearized terms of order p[t]^3

Out[30]= -((2*(2*Derivative[0, 1][a][t, y]^2 +
9*a[t, y]*Derivative[0, 2][a][t, y]))/(ep^2*p[t]^2)) +
(k*(8*q[t, y]*Derivative[0, 1][a][t, y]^2 +
17*a[t, y]*Derivative[0, 1][a][t, y]*Derivative[0, 1][q][t, y] +
36*a[t, y]*q[t, y]*Derivative[0, 2][a][t, y]))/(ep^3*
p[t]^3) + (-(1/(ep*p[t])) + (k*q[t, y])/(ep^2*p[t]^2))*
(ep*Derivative[2][p][
t] + (Derivative[1, 0][a][t,
y]*(ep*Derivative[1][p][t] + k*Derivative[1, 0][q][t, y]))/
a[t, y] +
k*Derivative[2, 0][q][t, y])[
ep*Derivative[1][p][t] + k*Derivative[1, 0][q][t, y], t]

Edit

I try to linearize this term in q[t] and p[t,y]

t00= (covd[Derivative[1][q][t] + Derivative[1, 0][p][t, y], t] +
2*(1 +
2*\[Alpha][t, r, \[Theta], \[Phi]])*(Derivative[0, 1][a][t, y]*
(4*(p[t, y] + q[t])*Derivative[0, 1][a][t, y] -  a[t, y]*Derivative[0, 1][p][t, y]) +
2*a[t, y]*(p[t, y] + q[t])*
Derivative[0, 2][a][t, y]))/(2*
a[t, y]^2*(p[t, y] + q[t])^3*  (1 + 2*\[Alpha][t, r, \[Theta], \[Phi]]))

by

Normal[Series[t00 /. {q -> (eps q[#] &), p -> (eps p[#] &)}, {eps, 0, 1}]]

But it does not work. (P[t]+q[t])^3 terms still there.

• In your expression ein11 there is an argument bracket ...[Derivative[1][p][t] + Derivative[1, 0][q][t, y], t] Is this intended or should it be a normal bracket? Nov 16, 2023 at 8:51
• Hi @UlrichNeumann. Thanks for your answer. May you please see the question edit. I try to use the answer's code in another new term, but it has been linearized. Nov 26, 2023 at 15:42
• In your code covd[] is undefined! Nov 26, 2023 at 17:10
• @UlrichNeumann. Yeah, it's a variable I will define later. Nov 26, 2023 at 17:14
• @UlrichNeumann. Also I found this replacement can help {q :> (eps q[##] &), p :> (eps p[##] &)}. Thanks for your reply. Nov 26, 2023 at 17:15

$$-\frac{2 \left(2 a^{(0,1)}(t,y)^2+9 a(t,y) a^{(0,2)}(t,y)\right)}{(p(t)+q(t))^2}$$