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## Solution of a first order ODE.

We have the first order ODE

$$y'=4t \sqrt y,~y(0)=1,$$

for which i have found the exact solution, namely a fourth order polynomial.

I want a numerical method to solve the problem exactly. This method has to be a fourth order method, since this implies that the local error vanishes.

Now we change the problem so it becomes

$$y'=4t \sqrt y - \lambda(y-(1+t^2)^2),~y(0)=a,$$

and the question is: for which values of $\lambda$ and $a$ does a method that has the above mentioned property solve the new problem exactly.

Of course, the obvious case is $\lambda=0$ and $a=1$, because in this case the new problem reduces to the first problem.

My idea is that the solution must be a fourth order polynomial, since a fourth order numerical method has to solve the new problem exactly.

Although I want your view on this and a strategy to find the values of $\lambda$ and $a$ for which the new problem is solved exactly by a fourth order numerical method.
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