Determining the sign of a solution to a Cauchy problemSuppose that y∈C1(R) is the solution of{y′(x)=sin⁡(y(x)+x2)x∈R,y(0)=0.

Step 1In a short description of the geometric situation, the functionf(x,y)=sin⁡(y+x2)has positive values f(x, 0) on the x-axis for x2∈(0,π). Thus on (−π, π) it can not be crossed downwards.The only way to have y(0)=0 is to have y&lt;0 for x&lt;0 and y&gt;0 for x&gt;0 inside this interval.For your task solution you might have to find more detailed justifications for these steps.

Expert answers to "Determining the sign of a solution to a..."

High School
Calculus and Analysis
Calculus 2
Differential equations

Leroy Davidson

Answered question

2022-03-27

Determining the sign of a solution to a Cauchy problem
Suppose that

y \in C^{1} (R)

is the solution of

{\begin{cases} y^{'} (x) = \sin (y (x) + x^{2}) x \in R, \\ y (0) = 0. \end{cases}

Answer & Explanation

Jamie Maldonado

Beginner2022-03-28Added 10 answers

Step 1
In a short description of the geometric situation, the function

f (x, y) = \sin (y + x^{2})

has positive values

f (x, 0)

on the x-axis for

x^{2} \in (0, π)

. Thus on

(- \sqrt{π}, \sqrt{π})

it can not be crossed downwards.
The only way to have

y (0) = 0

is to have

y < 0

for

x < 0

and

y > 0

for

x > 0

inside this interval.
For your task solution you might have to find more detailed justifications for these steps.

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