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Nasser
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The same system without the four initial conditions takes barely any time.

It has nothing to do with IC. It is FullSimplify which takes very long time. Try

Timing[DSolveValue[{OdeM, OdeP, Ic1, Ic2, Ic3, Ic4}, {V1[x], V2[x]}, 
  x]]

On my PC, V 13.2.1 it takes 4.5 seconds

Mathematica graphics

ps. I renamed your σX^2 to $w$. As I mentioned in your other question wrong-sign-in-variation-of-parameters-method , it is bad idea to use two letters for one variable. Why do you insist in using σX for a variable instead of say w or any other single letter?

Any way, it is well known that FullSimplify can slow things alot as it triedtries many types of simplifications. Try to use it only if needed. Simplify is much faster than FullSimplify

Full code

OdeM = 1/2*w^2*V1''[x] + ((a0*δ)/a1 - a1*x)*V1'[x] - 
    em*V1[x] == -((2*x - 2*ep*(a0 + a1*x)*λ)/(em - ep));
OdeP = 1/2*w^2*V2''[x] + ((a0*δ)/a1 - a1*x)*V2'[x] - 
    ep*V2[x] == -((-2*x + 2*em*(a0 + a1*x)*λ)/(em - ep));

Ic1 = em*V1[xhat] + ep*V2[xhat] == 0;
Ic2 = em*V1'[xhat] + ep*V2'[xhat] == 0;
Ic3 = V1[xhat] + V2[xhat] == (2*θ)/((ϵ - 1) (1 + φ))*xhat;
Ic4 = V1'[xhat] + V2'[xhat] == (2*θ)/((ϵ - 1) (1 + φ));
Timing[DSolveValue[{OdeM, OdeP, Ic1, Ic2, Ic3, Ic4}, {V1[x], V2[x]}, x]]

The same system without the four initial conditions takes barely any time.

It has nothing to do with IC. It is FullSimplify which takes very long time. Try

Timing[DSolveValue[{OdeM, OdeP, Ic1, Ic2, Ic3, Ic4}, {V1[x], V2[x]}, 
  x]]

On my PC, V 13.2.1 it takes 4.5 seconds

Mathematica graphics

ps. I renamed your σX^2 to $w$. As I mentioned in your other question wrong-sign-in-variation-of-parameters-method , it is bad idea to use two letters for one variable. Why do you insist in using σX for a variable instead of say w or any other single letter?

Any way, it is well known that FullSimplify can slow things alot as it tried many types of simplifications. Try to use it only if needed.

The same system without the four initial conditions takes barely any time.

It has nothing to do with IC. It is FullSimplify which takes very long time. Try

Timing[DSolveValue[{OdeM, OdeP, Ic1, Ic2, Ic3, Ic4}, {V1[x], V2[x]},x]]

On my PC, V 13.2.1 it takes 4.5 seconds

Mathematica graphics

ps. I renamed your σX^2 to $w$. As I mentioned in your other question wrong-sign-in-variation-of-parameters-method , it is bad idea to use two letters for one variable. Why do you insist in using σX for a variable instead of say w or any other single letter?

Any way, it is well known that FullSimplify can slow things alot as it tries many types of simplifications. Try to use it only if needed. Simplify is much faster than FullSimplify

Full code

OdeM = 1/2*w^2*V1''[x] + ((a0*δ)/a1 - a1*x)*V1'[x] - 
    em*V1[x] == -((2*x - 2*ep*(a0 + a1*x)*λ)/(em - ep));
OdeP = 1/2*w^2*V2''[x] + ((a0*δ)/a1 - a1*x)*V2'[x] - 
    ep*V2[x] == -((-2*x + 2*em*(a0 + a1*x)*λ)/(em - ep));

Ic1 = em*V1[xhat] + ep*V2[xhat] == 0;
Ic2 = em*V1'[xhat] + ep*V2'[xhat] == 0;
Ic3 = V1[xhat] + V2[xhat] == (2*θ)/((ϵ - 1) (1 + φ))*xhat;
Ic4 = V1'[xhat] + V2'[xhat] == (2*θ)/((ϵ - 1) (1 + φ));
Timing[DSolveValue[{OdeM, OdeP, Ic1, Ic2, Ic3, Ic4}, {V1[x], V2[x]}, x]]
Source Link
Nasser
  • 150.6k
  • 12
  • 162
  • 376

The same system without the four initial conditions takes barely any time.

It has nothing to do with IC. It is FullSimplify which takes very long time. Try

Timing[DSolveValue[{OdeM, OdeP, Ic1, Ic2, Ic3, Ic4}, {V1[x], V2[x]}, 
  x]]

On my PC, V 13.2.1 it takes 4.5 seconds

Mathematica graphics

ps. I renamed your σX^2 to $w$. As I mentioned in your other question wrong-sign-in-variation-of-parameters-method , it is bad idea to use two letters for one variable. Why do you insist in using σX for a variable instead of say w or any other single letter?

Any way, it is well known that FullSimplify can slow things alot as it tried many types of simplifications. Try to use it only if needed.