Problems

01

Hopping from the $x$-axis

Consider the IVP: $y^{\prime }(t)=3y^{2/3},y(0)=0$. Analyze the functional solutions to this IVP. Extend your analysis to $t<0$ as well. This Desmos and picture might help you:
https://www.desmos.com/calculator/n8yfpnshxd

02

When does Picard-Lindelöf apply?

Which of the following IVPs satisfy the conditions of the Picard-Lindelöf theorem?

• (a) $y^{\prime }=\sqrt{y},y(0)=1$
• (b) $y^{\prime }=1+y^2,y(0)=0$
• (c) $y^{\prime }=\frac{t-y}{t+y},y(1)=-1$
• (d) $y^{\prime }=\cos (t+y),y(t_0)=y_0$

03

Understanding existence and uniqueness

Consider the IVP of the previous problem, part (c):

$$y^{\prime }=\frac{t-y}{t+y},y(1)=-1.$$

• (a) Precisely which initial data $(t_0,y_0)$ are such that the Picard-Lindelöf theorem is satisfied, guaranteeing existence and uniqueness of local solutions? Which would ensure failure? (Your answer would also answer Problem 04-02(c) as a specific case.)

• (b) Solve the homogeneous differential equation very carefully (double check your solution set). Exactly which solutions blow up? (For this question, specify the solutions using initial conditions living either on the $y$-axis or on the $x$-axis.)

For both parts of this problem, you should graph the slope field using Desmos: https://www.desmos.com/calculator/gypelekic6. For part (b), you should graph your solutions on top of the slope field. It is recommended to implement your constant as a slider so you can move the solution curves around. (No need to submit your graphs.)

04

Picard iterates

• (a) Compute the Picard iterates up to $y_3$, starting with $y_0=0$, for the IVP: $$y^{\prime }=t+y^2,y(0)=0.$$
• (b) Compute the Picard iterates up to $y_4$, starting with $y_0=0$, for the IVP: $$y^{\prime }=1+(y-t{)}^2,y(0)=0.$$