# how to write to solve this type of system of equations ?

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NIRUPAM SAHOO on 11 Sep 2021
Commented: NIRUPAM SAHOO on 12 Sep 2021
NIRUPAM SAHOO on 12 Sep 2021
please anyone solve this . here u and v are functions of r.

Wan Ji on 12 Sep 2021
Hey friend
Just expand the left items of the two equations, extract u'' and v'', then an ode45 solver is there for you.

Walter Roberson on 12 Sep 2021
Edited: Walter Roberson on 12 Sep 2021
I was not able to figure out what is being raised to 10/9 . I used squiggle instead.
syms u(r) v(r)
syms N squiggle real
assume(r, 'real')
du = diff(u);
dv = diff(v);
left1 = diff(r^(N-1)*du^3)
left1(r) =
right1 = r^(N-1) * sqrt(u) * sqrt(v) / (3*r^(2/3) * sqrt(1 + 9*squiggle^(10/9)/(10*(3*N-2)^(1/3))))
right1(r) =
left2 = diff(r^(N-1)*dv^3)
left2(r) =
right2 = r^(N-1) * u * v / (3*r^(2/3))
right2(r) =
eqn1 = left1 == right1
eqn1(r) =
eqn2 = left2 == right2
eqn2(r) =
ic = [u(0) == 1, v(0) == 1, du(0) == 0, dv(0) == 0]
ic =
sol = dsolve([eqn1, eqn2, ic])
Warning: Unable to find symbolic solution.
sol = [ empty sym ]
string(eqn1)
ans = "3*r^(N - 1)*diff(u(r), r)^2*diff(u(r), r, r) + r^(N - 2)*(N - 1)*diff(u(r), r)^3 == (r^(N - 1)*u(r)^(1/2)*v(r)^(1/2))/(3*r^(2/3)*((9*squiggle^(10/9))/(10*(3*N - 2)^(1/3)) + 1)^(1/2))"
string(eqn2)
ans = "3*r^(N - 1)*diff(v(r), r)^2*diff(v(r), r, r) + r^(N - 2)*(N - 1)*diff(v(r), r)^3 == (r^(N - 1)*u(r)*v(r))/(3*r^(2/3))"
string(ic)
ans = 1×4 string array
"u(0) == 1" "v(0) == 1" "subs(diff(u(r), r), r, 0) == 0" "subs(diff(v(r), r), r, 0) == 0"
Lack of a symbolic solution means that you would have to do numeric solutions -- but you cannot do a numeric solution to infinity, and you certainly would not get a formula out of it.
NIRUPAM SAHOO on 12 Sep 2021
syms p(t) m(t) t Y
Eqns = [diff((t^(100-1))*(diff(p(t),t))) == (t^(100-1))*(t-1)*exp(t)*p(t)*m(t);
diff((t^(100-1))*(diff(m(t),t))) == (t^(100-1))*(t-1)*exp(t)*p(t)^(1/2)*m(t)^(1/2)]
[DEsys,Subs] = odeToVectorField(Eqns);
DEFcn = matlabFunction(DEsys, 'Vars',{t,Y});
tspan = [0,100];
y0 = [0 0 0 0];
[t,Y] = ode45(DEFcn, tspan, y0);
plot(t,Y)

R2021a

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