<p>I don't think your reasoning is correct. Some remarks:</p>
<ul>
<li><p>If you have an optimization problem, then you assume that you know the objective function, which in your case contains the function <span class="math-container">$u(x,y)$</span>. As such, this function is known and you want to find the particular values <span class="math-container">$x^*$</span> and <span class="math-container">$y^*$</span> that maximizes your objective function. In other words, the optimal values <span class="math-container">$x^*$</span> and <span class="math-container">$y^*$</span> are unknowns but <span class="math-container">$u(x,y)$</span> itself is known. If you solve an ode, then the function itself is unknown.</p>
</li>
<li><p>Next, notice that if you have an ode, like
<span class="math-container">$$
u_x(3u(x,y)2x−3(1−u(x,y))2y)+u(x,y)3=0
$$</span>
Then you inherently assume that this equation holds for <strong>all</strong> values of <span class="math-container">$x$</span> and <span class="math-container">$y$</span> in the domain of <span class="math-container">$u$</span>. In other words, you have an identity. <a href="https://en.wikipedia.org/wiki/Identity_(mathematics)" rel="nofollow noreferrer">wiki link</a></p>
</li>
<li><p>On the other hand, if you derive the first order conditions for an optimization problem, then this first order condition will only hold at the particular (candidate) optimal points <span class="math-container">$x^\ast$</span> and <span class="math-container">$y^\ast$</span> but you will not get an identity that holds for all <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. As such, the methods used to solve ODE's are not suitable to solve first order conditions.</p>
</li>
</ul>

CC BY-SA 4.0
2022-05-19T23:52:03.627Z
Thanks for your answer! I am not maximizing for x and y. Some background to my problem - agent is trying signal x and y to the society through function u and he is trying to choose that x and y which will benefit him the most
Elina Gilbert
CC BY-SA 4.0
2022-05-21T07:57:15.517Z
I don't get it: If he chooses $x$ and $y$ that benefits him the most then he is indeed maximising for $x$ and $y$.
tdm
CC BY-SA 4.0
2022-05-22T08:12:13.297Z
He is maximizing with $x$ and $y$, not $for$ $x$ and $y$
Elina Gilbert
CC BY-SA 4.0
2022-05-19T17:38:50.973Z
2022-05-19T17:38:50.973Z
tdm
<p>I have following function -</p>
<p><span class="math-container">$$ \max_{x, y} ~ u(x, y)^{3}x + (1-u(x, y))^{3}y$$</span></p>
<p>FOC:
<span class="math-container">$$u_{x}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +u(x,y)^{3} = 0$$</span>... (1)</p>
<p><span class="math-container">$$u_{y}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +(1-u(x,y))^{3} = 0$$</span>... (2)</p>
<p>Can I solve it like -
Treat (1) as ode in x and (2) as ode in y  and solve them separately and then get u(x, y) = <span class="math-container">$sol_{(1)}*sol_{(2)}$</span></p>
<p>I just need clarification and guidance if I am wrong.</p>
<p>Thank you</p>

<optimization><difference-equations>
System of first order partial differential equation
Elina Gilbert
# System of first order partial differential equation

(Asked by: username_0 on 2022-05-19)

I have following function -

$$ \max\_{x, y} ~ u(x, y)^{3}x + (1-u(x, y))^{3}y$$

FOC:
$$u\_{x}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +u(x,y)^{3} = 0$$... (1)

$$u\_{y}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +(1-u(x,y))^{3} = 0$$... (2)

Can I solve it like -
Treat (1) as ode in x and (2) as ode in y and solve them separately and then get u(x, y) = $sol\_{(1)}\*sol\_{(2)}$

I just need clarification and guidance if I am wrong.

Thank you

**Tags:** optimization, difference-equations


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## 1 answers

### Answer by username_1 on 2022-05-19. Score: 2

I don't think your reasoning is correct. Some remarks:

* If you have an optimization problem, then you assume that you know the objective function, which in your case contains the function $u(x,y)$. As such, this function is known and you want to find the particular values $x^\*$ and $y^\*$ that maximizes your objective function. In other words, the optimal values $x^\*$ and $y^\*$ are unknowns but $u(x,y)$ itself is known. If you solve an ode, then the function itself is unknown.
* Next, notice that if you have an ode, like
  $$
  u\_x(3u(x,y)2x−3(1−u(x,y))2y)+u(x,y)3=0
  $$
  Then you inherently assume that this equation holds for **all** values of $x$ and $y$ in the domain of $u$. In other words, you have an identity. [wiki link](https://en.wikipedia.org/wiki/Identity_(mathematics))
* On the other hand, if you derive the first order conditions for an optimization problem, then this first order condition will only hold at the particular (candidate) optimal points $x^\ast$ and $y^\ast$ but you will not get an identity that holds for all $x$ and $y$. As such, the methods used to solve ODE's are not suitable to solve first order conditions.

> **Comments:**
> **username_0**: Thanks for your answer! I am not maximizing for x and y. Some background to my problem - agent is trying signal x and y to the society through function u and he is trying to choose that x and y which will benefit him the most  
> **username_1**: I don't get it: If he chooses $x$ and $y$ that benefits him the most then he is indeed maximising for $x$ and $y$.  
> **username_0**: He is maximizing with $x$ and $y$, not $for$ $x$ and $y$  

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