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Question: dsolve gives solution to second order order with one IC but missing constant of integration.

Question:dsolve gives solution to second order order with one IC but missing constant of integration.

nm 11353 Maple 2024
ode dsolve differential-equations
August 25 2024
2

At school the teacher always said that if we have second order ode and only one initial conditions (say y'(0)=0 or y(0)=0) then the solution should have one constant of integration in it.

And if we have no initial conditions, then the solution should have 2 constants of integrations in it.

And if we have two initial conditions, then the solution should have zero constants of integrations in it.

In this example, Maple is given second order ode with one IC. But the solution it gives when asked to solve it explicit, has no constant of integration in it at all. 

When asked to solve it using implicit, then the constant of integration shows up. 

Both solutions actually verify to be fully correct using odetest. So it looks like the solution as explicit is particular solution and not a general solution.

Why is that? Why it did not give general solution when asked to solve the ode as explicit?

> 

restart;
> 

interface(version);

`Standard Worksheet Interface, Maple 2024.1, Windows 10, June 25 2024 Build ID 1835466`

> 

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1793 and is the same as the version installed in this computer, created 2024, August 25, 9:6 hours Pacific Time.`

> 

libname;

"C:\Users\Owner\maple\toolbox\2024\Physics Updates\lib", "C:\Program Files\Maple 2024\lib"

> 

restart;
> 

ode:=y(x)*diff(y(x),x$2)+diff(y(x),x)^2+1=0;
IC:=D(y)(0)=1;

y(x)*(diff(diff(y(x), x), x))+(diff(y(x), x))^2+1 = 0

(D(y))(0) = 1

> 

sol1:=dsolve([ode,IC],explicit);
 

y(x) = (-x^2+x*2^(1/2)+1/2)^(1/2)

> 

sol2:=dsolve([ode,IC],implicit);

-(1/2)*y(x)^2+x*y(0)-(1/2)*x^2+c__2 = 0

> 

odetest(sol1,[ode,IC]);
odetest(sol2,[ode,IC]);

[0, 0]

[0, 0]

 

 

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