Can 2 different ODE's have the same set of solutions?If I have two differents linear ODE's:x''+p(t)x'+q(t)x=f(t).&nbsp;p,q∈C(I,∞).x''+j(t)x'+g(t)x=h(t).&nbsp;j,g∈C(I,∞).Coul they have exactly the same set of solutions?And if we have two different non-linear ODE's could they?

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Can 2 different ODE&#039;s have the same set of solutions?If I have two differents linear ODE&#039;s:x&#039;&#039;+p(t)x&#039;+q(t)x=f(t).&amp;nbsp;p,q∈C(I,∞).x&#039;&#039;+j(t)x&#039;+g(t)x=h(t).&amp;nbsp;j,g∈C(I,∞).Coul they have exactly the same set of solutions?And if we have two different non-linear ODE&#039;s could they?

pintorreeqwf · Accepted Answer

Step 1If your ODE&#039;s are x(n)=F(t,x,x′,⋯,x(n−1)) and G(t,x,x′,⋯,x(n−1)), for some F,G:Rn+1→R and they both have local solutions for any initial conditions, then we must have F=G: if not, just take a value v=(t0,x0,x0′,⋯,x0(n−1)) such that F(v)≠G(v) and then your local solution at v will be different in both equations, as both solutions will have different n-th derivatives.Step 2In the linear case, for the equations you give we would have the functions F(t,x,x′)=−p(t)x′−q(t)x−f(t) and G(t,x,x′)=−j(t)x′−g(t)x−h(t). So if they have the same solutions, we need F(t,x,x′)=G(t,x,x′);∀t,x,x′, which implies p(t)=j(t),q(t)=g(t),f(t)=h(t): if for example p(t0)≠j(t0) for some t0, then F(t0,0,s)=−sp(t0)−f(t0) and G(t0,0,s)=−sj(t0)−h(t0), so F(t0,0,s) cannot be equal to G(t0,0,s);∀s, contradicting F=G. Similar arguments can be used to show q(t)=g(t) and f(t)=h(t).

Can 2 different ODE's have the same set

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