Maximum value a function satisfying a differential equation can achieve.

• MHB
• caffeinemachine
Then I ran this code in Octave:function Ydot = g(Y, x) u = Y(1); y = Y(2); Ydot = [-y - x*abs(sin(x))*u, u];endfunctiont = linspace(0, 10, 100);Y0 = [4, -3];[Y, t] = lsode("g", Y0, t);disp([t, Y])In summary, we are asked to find the maximum value of a twice-differentiable function $f(x)$ on the positive real line, given that $f(x)+f^{\prime\prime}(x)=-x|\sin(x)|f'( caffeinemachine Gold Member MHB Let$f:\mathbb R\to \mathbb R$be a twice-differentiable function such that$f(x)+f^{\prime\prime}(x)=-x|\sin(x)|f'(x)$for$x\geq 0$. Assume that$f(0)=-3$and$f'(0)=4$. Then what is the maximum value that$f$achieves on the positive real line? a) 4 b) 3 c) 5 d) Maximum value does not exist. I am quite lost on this one. After some thought I am convinced that the maximum value should exist, though I do not have a good argument to support this claim. When I throw it at Octave online, I get: View attachment 9668 So it seems it is none of the above, but the answer$3$is close. For reference, I have re-encoded the differential equation a bit to solve it with Octave's [M]lsode[/M]. Let$y=f(x)$,$u=y'$, and$Y=(u,y)$then:$f(x)+f''(x)=-x|\sin x|f'(x) \\
y + u' = -x|\sin x|u \\
\begin{cases}u'=-y-x|\sin x|u\\ y'=u\end{cases}\\
(u', y') = (-y-x|\sin x|u,\, u) =: g(u,y;x)\\
Y' = g(Y;x)
\$

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What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model many physical and natural phenomena in fields such as physics, engineering, and biology.

What is the maximum value a function can achieve?

The maximum value a function can achieve is the highest point on the graph of the function. It represents the largest output value that the function can produce for a given input.

How is the maximum value of a function related to its derivative?

The maximum value of a function is related to its derivative through the critical points of the function. These are the points where the derivative is equal to 0 or undefined, and they can help determine the maximum value of the function.

Can a function have multiple maximum values?

Yes, a function can have multiple maximum values if it is not a strictly increasing or decreasing function. In such cases, there may be several local maximum values, but only one global maximum value.

How can I find the maximum value of a function satisfying a differential equation?

To find the maximum value of a function satisfying a differential equation, you can use techniques such as taking derivatives, setting the derivative equal to 0, and solving for critical points. You can also use graphical methods or numerical methods such as optimization algorithms to find the maximum value.

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