解方程(二）

northwolves

$cos^7x+1/(cos^3x)=sin^7x+1/(sin^3x)$

lsr314

显然$x=pi/4+k pi$是其解，从图像上看这是全部实数解。

wayne

根据$f(t)=t^7+1/t^3$的奇偶性,容易看出周期性是$\pi$,且咱们只需要考虑$[0,\pi/2]$区间，又考虑到方程是关于$sin(x)$与$cos(x)$对称的，即关于$x=\frac{\pi}{4}$对称。咱们只需要考虑$[0,\pi/4]$，而$f(t)=t^7+1/t^3$在$0<t<\frac{\sqrt{2}}{2}$是单调递减的，所以只可能$sin(x)=cos(x)$，于是答案就是$x=\frac{\pi}{4}+k\pi$

mathematica

1. (*https://bbs.emath.ac.cn/thread-17673-1-1.html解方程(二）*)
   
2. Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
   
3. ans=NSolve[{x^7+1/x^3==y^7+1/y^3,x^2+y^2==1},{x,y}];
   
4. Grid[ans,Alignment->Left]
   

复制代码

唯一正确的标准答案来了！人家又没说是实数范围内！
\[\begin{array}{ll}
x\to 1.17636\, -0.940122 i & y\to -1.17636-0.940122 i \\
x\to 1.17636\, +0.940122 i & y\to -1.17636+0.940122 i \\
x\to -1.17636-0.940122 i & y\to 1.17636\, -0.940122 i \\
x\to -1.17636+0.940122 i & y\to 1.17636\, +0.940122 i \\
x\to 1.06913\, +0.3456 i & y\to 0.598232\, -0.617638 i \\
x\to 1.06913\, -0.3456 i & y\to 0.598232\, +0.617638 i \\
x\to -1.06913+0.3456 i & y\to -0.598232-0.617638 i \\
x\to -1.06913-0.3456 i & y\to -0.598232+0.617638 i \\
x\to 0.598232\, -0.617638 i & y\to 1.06913\, +0.3456 i \\
x\to 0.598232\, +0.617638 i & y\to 1.06913\, -0.3456 i \\
x\to -0.598232-0.617638 i & y\to -1.06913+0.3456 i \\
x\to -0.598232+0.617638 i & y\to -1.06913-0.3456 i \\
x\to -0.707107 & y\to -0.707107 \\
x\to 0.707107 & y\to 0.707107 \\
x\to -1.16688-0.119569 i & y\to 0.221599\, -0.629617 i \\
x\to -1.16688+0.119569 i & y\to 0.221599\, +0.629617 i \\
x\to 1.16688\, -0.119569 i & y\to -0.221599-0.629617 i \\
x\to 1.16688\, +0.119569 i & y\to -0.221599+0.629617 i \\
x\to 0.221599\, -0.629617 i & y\to -1.16688-0.119569 i \\
x\to 0.221599\, +0.629617 i & y\to -1.16688+0.119569 i \\
x\to -0.221599-0.629617 i & y\to 1.16688\, -0.119569 i \\
x\to -0.221599+0.629617 i & y\to 1.16688\, +0.119569 i \\
x\to 0.797089\, -0.367901 i & y\to -0.797089-0.367901 i \\
x\to 0.797089\, +0.367901 i & y\to -0.797089+0.367901 i \\
x\to -0.797089-0.367901 i & y\to 0.797089\, -0.367901 i \\
x\to -0.797089+0.367901 i & y\to 0.797089\, +0.367901 i \\
\end{array}\]

mathematica

老问题
https://bbs.emath.ac.cn/forum.ph ... 290&fromuid=865