Checking the solution of a differential equation with a given solution

Question

I can't verify the solution of this differential equation.

 eqn = y'[x] == x*y[x]^(1/2)

where:

sol = {{y -> x^4/16}}

When I tried the ReplaceAll statement (/.) to verify the solution I get:

In[6]:= eqn /. sol

Out[6]= {Derivative[1][(x^4/16)][x] == x Sqrt[((x^4)/16)[x]]}

Instead of {True}. I have checked on:

• Checking differential equation solution

• https://reference.wolfram.com/language/tutorial/DSolveSolutionVerification.html

And I can't figure out what I'm doing wrong. I'm just looking for some pointers or tips to see what I can do.

Answer 1

To verify ODE solution, use the functional method, as described in tutorial/DSolveSolutionVerification.html i.e. use y and not y[x] in the DSolve command.

Sometimes, as in this case, you have to help it a little also using Simplify as mentioned in the above link.

For your example:

ClearAll[y, x]
ode = y'[x] == x Sqrt[y[x]];
sol = DSolve[ode, y, x]
Simplify[eq /. sol, 
 Assumptions -> Element[{x, C[1]}, Reals] && x > 0 && C[1] > 0]

If you do not use the assumptions, then Mathematica does not know that the sqrt is real or complex. Here is without the assumptions:

ClearAll[y, x]
ode = y'[x] == x Sqrt[y[x]];
sol = DSolve[ode, y, x]
Simplify[eq /. sol]

btw, I always wished Mathematica has an odetest() function as in Maple, as it makes it little easier. This is the same thing in Maple

ode := diff(y(x),x)= x*sqrt(y(x));
sol:=dsolve(ode,y(x));
odetest(sol,ode);
           0

If the answer is zero, then the solution satisfies the ODE.

ps. I just saw your edit and you wanted to verify x^4/16. Ok, in this case, you write this:

ode = y'[x] == x Sqrt[y[x]];
sol2 = y -> Function[{x}, x^4/16];
Simplify[eq /. sol2, Assumptions -> x > 0]

The idea is to use Function, so that it will work with derivatives.