Questions

01

Checking things

• For the first-order equation above, why must $F(x)+c$ give a complete set of solutions?
• For the second-order equation, explain how to obtain the formula $\Phi (x)+c_{1}x+c_2$.

02

Higher order

(Big Bonus question. Not for the faint of heart!)

• Can any two points $(x_0,y_0)$ and $(x_1,y_1)$ be chosen?
• Now iterate the process: consider $y^{(n)}=f(x)$ and solve using multiple consecutive integrations. Describe the family of solutions. What constraints are there on the set of points $(x_0,y_0),\dots ,(x_n,y_n)$ needed to specify a particular solution?