2. Discuss the difference between the differential equations $$ \left(y^{\prime}\right)^{2}=4 y \quad \text { and } \quad y^{\prime}=2 \sqrt{y} $$ Do they have the same solution curves? Why or why not? Determine the points $$ \left(x_{0}, y_{0}\right) $$ in the plane for which the initial value problem $$ y^{\prime}=2 \sqrt{y}, y\left(x_{0}\right)=y_{0} $$ has (a) no solutions, (b) a unique solution, or (c) infinitely many solutions.

Question

UpStudy AI · Accepted Answer

To discuss the difference between the differential equations $$ \left(y^{\prime}\right)^{2}=4 y \quad \text{ and } \quad y^{\prime}=2 \sqrt{y}, $$ we first note that the first equation can be rewritten as: $$ y' = \pm 2\sqrt{y}. $$ This indicates that the first equation has two possible forms for $$y'$$: $$y' = 2\sqrt{y}$$ and $$y' = -2\sqrt{y}$$. ### Comparison of Solution Curves 1. **Same Solution Curves**: - The equation $$y' = 2\sqrt{y}$$ describes the behavior of $$y$$ increasing with respect to $$x$$ when $$y$$ is non-negative. - The equation $$y' = -2\sqrt{y}$$ describes the behavior of $$y$$ decreasing with respect to $$x$$ when $$y$$ is non-negative. - Therefore, the two equations do not have the same solution curves because the first equation allows for both increasing and decreasing solutions, while the second only allows for increasing solutions. ### Initial Value Problem Analysis Now, let's determine the points $$(x_0, y_0)$$ in the plane for which the initial value problem $$y' = 2\sqrt{y}, y(x_0) = y_0$$ has: (a) **No Solutions**: - If $$y_0 < 0$$, then there are no solutions because the square root function is not defined for negative values. Thus, the initial value problem has no solutions for any $$x_0$$ when $$y_0 < 0$$. (b) **Unique Solution**: - If $$y_0 = 0$$, the equation becomes $$y' = 2\sqrt{0} = 0$$. The only solution is the constant solution $$y(x) = 0$$ for all $$x$$, which is unique. - If $$y_0 > 0$$, the equation $$y' = 2\sqrt{y}$$ can be solved uniquely using separation of variables, leading to a unique solution for each positive initial condition. (c) **Infinitely Many Solutions**: - The case where $$y_0 = 0$$ also leads to infinitely many solutions if we consider the general solution of the first equation $$y' = -2\sqrt{y}$$ because it can lead to a family of solutions that converge to the line $$y = 0$$ from above. However, for the specific equation $$y' = 2\sqrt{y}$$, there are no infinitely many solutions. ### Summary - **No Solutions**: $$y_0 < 0$$ - **Unique Solution**: $$y_0 = 0$$ or $$y_0 > 0$$ - **Infinitely Many Solutions**: Not applicable for $$y' = 2\sqrt{y}$$ specifically, but can be considered in the context of the general solution of the first equation. Thus, the two differential equations do not have the same solution curves due to the presence of both increasing and decreasing solutions in the first equation, while the second equation only allows for increasing solutions. The two differential equations do not have the same solution curves because one allows both increasing and decreasing solutions, while the other only allows increasing solutions. For the equation $$ y' = 2\sqrt{y} $$: - **No Solutions**: When $$ y_0 < 0 $$ (since the square root of a negative number is not real). - **Unique Solution**: When $$ y_0 = 0 $$ or $$ y_0 > 0 $$. - **Infinitely Many Solutions**: Not applicable for this specific equation. Therefore, the initial value problem $$ y' = 2\sqrt{y} $$ has: - No solutions when $$ y_0 < 0 $$. - A unique solution when $$ y_0 = 0 $$ or $$ y_0 > 0 $$. - No infinitely many solutions in this context.

Discuss the difference between the differential equations

Do they have the same solution curves? Why or why not? Determine the points in
the plane for which the initial value problem has (a) no solutions, (b) a
unique solution, or © infinitely many solutions.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Comparison of Solution Curves

Initial Value Problem Analysis

Summary

Mind Expander

preguntas relacionadas

Latest Calculus Questions

Discuss the difference between the differential equations Do they have the same solution curves? Why or why not? Determine the points in the plane for which the initial value problem has (a) no solutions, (b) a unique solution, or © infinitely many solutions.