For which real values of \(p\) does the differential equation \[ y''+2py'+y=3 \] admit a solution with infinitely many critical points?
Solution
1. For every real value of \(p\), the constant function\[ y(x)\equiv3 \]is a solution of the differential equation.
For the constant function \(y=3\), \[ y'=0, \qquad y''=0. \] Hence, for every \(p\in\mathbb R\), \[ y''+2py'+y = 0+0+3 =3. \]
2. This solution has infinitely many critical points.
A critical point of a differentiable real-valued function is a point where its derivative vanishes. For the solution in step 1, \[ y'(x)=0 \] for every \(x\in\mathbb R\). Thus every real number is a critical point.
3. The required set of parameters is\[ \boxed{\mathbb R}. \]
Steps 1 and 2 exhibit, for every real \(p\), a solution with infinitely many critical points. There are no other real parameter values to consider.
4. Q.E.D.
Step 3 is the requested classification of the real parameters.