Homework Sourseto solve the following nonlinear equations yex and yx1x Defi
to solve the following nonlinear equations :y=e^x and y=x(1+x)
Define an objective function that can be maximized to obtain a solution to these
equations.
Solution
Well if we want to find a solution to the equations you have given then let the objective function be
\\begin{align} F(x) = -|f_1(x)-f_2(x)| \\end{align}
where $f_1(x) = e^x$ and $f_2(x)= x(x+1)$. Then maximising $F$ is equivalent to minimising the distance between the two functions which will yield a solution to the system. You can quickly sketch the function in question by first sketching the function inside the absolute value sign and then reflecting about the x-axis whenever it goes below zero. Then \"flip\" it and you should get an idea where the solution is (the maximum of the resulting function).
| Well if we want to find a solution to the equations you have given then let the objective function be \\begin{align} F(x) = -|f_1(x)-f_2(x)| \\end{align} where $f_1(x) = e^x$ and $f_2(x)= x(x+1)$. Then maximising $F$ is equivalent to minimising the distance between the two functions which will yield a solution to the system. You can quickly sketch the function in question by first sketching the function inside the absolute value sign and then reflecting about the x-axis whenever it goes below zero. Then \"flip\" it and you should get an idea where the solution is (the maximum of the resulting function). |
