Definitions: Question 1-1.5 & Solution

Question 5

Consider the initial value problem $y'=2\sqrt{y},\qquad y(0)=0,$ where the square root is the nonnegative real root. For each real parameter $a\ge 0$ , define on $\mathbb R$ $y_a(x)=\begin{cases} 0,&x\le a,\\ (x-a)^2,&x>a. \end{cases}$ Tasks

Prove that $y_a$ is continuously differentiable at its joining point $x=a$ .
Verify the differential equation everywhere, including at $x=a$ , and verify the initial value.
Show that different parameters give distinct solutions of the same IVP. Identify an additional solution that never leaves $0$ .
Sketch $y_0$ , $y_1$ , and $y_2$ in your solution and explain what this example shows about uniqueness. Does it mean that every first-order IVP is nonunique?

Original worksheet page 1: question and worked solution for 1-1-005

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Question 5 – Solution

Strategy. Check the joining point directly; solving on the two sides alone is insufficient. Then compare solutions with the same initial value.

Step 1: Smooth joining. Both pieces tend to $0$ as $x\to a$ . At $a$ , the difference quotient is $0$ for $h<0$ and $h^2/h=h$ for $h>0$ . Hence $y_a'(a)=0$ and $y_a'(x)=\begin{cases}0,&x\le a,\\2(x-a),&x>a.\end{cases}$ This derivative is continuous at $a$ , so $y_a$ is $C^1$ on $\mathbb R$ .

Step 2: Equation and initial value. For $x\le a$ , both $y_a'$ and $2\sqrt{y_a}$ are $0$ . For $x>a$ , $x-a>0$ , so $2\sqrt{(x-a)^2}=2|x-a|=2(x-a)=y_a'.$ Since $a\ge 0$ , $y_a(0)=0$ . Thus

See the diagram in the original worksheet below.

Step 3: Distinctness. If $0\le a<b$ , choose $a<x<b$ . Then $y_a(x)>0$ but $y_b(x)=0$ . Also $y\equiv 0$ is a solution distinct from every finite-parameter $y_a$ .

Step 4: Meaning. This IVP has infinitely many solutions: the initial value does not determine when a solution leaves $0$ . It disproves automatic uniqueness for all first-order IVPs; it does not assert nonuniqueness for every IVP. No general uniqueness theorem is needed for this counterexample.

Original worksheet page 2: question and worked solution for 1-1-005

Definitions — Question 5

Question 5

Question 5 – Solution