方程 x · cosx = 42 是否有实数解？

dodonaomikiki

联想：  \(       42=x  \bullet  cosx          \)   不晓得能否得到解答？

题目如题！

由一个参数方程联想得来！

xfhaoym

xcosx的最大值也就是约为0.5611。

luyuanhong

问  方程 x · cosx = 42 是否有实数解？

答  有。例如，有一个实数解是 x=43.7022718… 。

王守恩

主帖在问什么——我看不懂。围绕 42=x*cosx  我来发挥一下。

代码 1——NSolve[{42 == x*Cos[x], 10*Pi + 42 > x > 0}, {x}]—— x 可以有无穷个解。——省略了。
{{x -> 43.7023}, {x -> 44.3064}, {x -> 49.7013}, {x -> 50.8648}, {x ->55.8294}, {x -> 57.2967}, {x -> 62.0052}, {x -> 63.6825}, {x -> 68.2076}, {x -> 70.0428}}

代码 2——代码 1—42 = x*Cos[x] 与 42 = x*Cos[x + 2 k*Pi]—代码 2——解是相通的·。
代码 2——Table[NSolve[{42 == x*Cos[x + 2 k*Pi], 10 \[Pi] + 42 > x > 0}, {x}], {k, -3, 3}]
{{x -> 43.7023}, {x -> 44.3064}, {x -> 49.7013}, {x -> 50.8648}, {x -> 55.8294}, {x -> 57.2967}, {x -> 62.0052}, {x -> 63.6825}, {x -> 68.2076}, {x -> 70.0428}},
{{x -> 43.7023}, {x -> 44.3064}, {x -> 49.7013}, {x -> 50.8648}, {x -> 55.8294}, {x -> 57.2967}, {x -> 62.0052}, {x -> 63.6825}, {x -> 68.2076}, {x -> 70.0428}},
{{x -> 43.7023}, {x -> 44.3064}, {x -> 49.7013}, {x -> 50.8648}, {x -> 55.8294}, {x -> 57.2967}, {x -> 62.0052}, {x -> 63.6825}, {x -> 68.2076}, {x -> 70.0428}},
{{x -> 43.7023}, {x -> 44.3064}, {x -> 49.7013}, {x -> 50.8648}, {x -> 55.8294}, {x -> 57.2967}, {x -> 62.0052}, {x -> 63.6825}, {x -> 68.2076}, {x -> 70.0428}},
{{x -> 43.7023}, {x -> 44.3064}, {x -> 49.7013}, {x -> 50.8648}, {x -> 55.8294}, {x -> 57.2967}, {x -> 62.0052}, {x -> 63.6825}, {x -> 68.2076}, {x -> 70.0428}},
{{x -> 43.7023}, {x -> 44.3064}, {x -> 49.7013}, {x -> 50.8648}, {x -> 55.8294}, {x -> 57.2967}, {x -> 62.0052}, {x -> 63.6825}, {x -> 68.2076}, {x -> 70.0428}},
{{x -> 43.7023}, {x -> 44.3064}, {x -> 49.7013}, {x -> 50.8648}, {x -> 55.8294}, {x -> 57.2967}, {x -> 62.0052}, {x -> 63.6825}, {x -> 68.2076}, {x -> 70.0428}}}

x 有无穷个解——其中有一个是最小的——我们只能关心最小的那个解。

代码 3——把n=0—42的最小解都列出来。
代码 3——Table[{n, x /. First@NSolve[{n ==x*Cos[x], 6 + n > x > n}, {x}]}, {n, 0, 42}]
{{0, 1.5708}, {1, 4.91719}, {2, 5.11418}, {3, 5.31247}, {4, 5.52243}, {5, 5.76281}, {6, 6.10099}, {7, 11.6407}, {8, 11.7449}, {9, 11.8574}, {10, 11.9828}, {11, 12.1311},
{12, 12.3333}, {13, 18.0811}, {14, 18.1591}, {15, 18.244}, {16, 18.339}, {17, 18.4504}, {18, 18.5957}, {19, 24.4519}, {20, 24.516}, {21, 24.5859}, {22, 24.6637}, {23, 24.754},
{24, 24.8678}, {25, 25.0622}, {26, 30.8477}, {27, 30.9076}, {28, 30.9741}, {29, 31.0505}, {30, 31.144}, {31, 31.2816}, {32, 37.1656}, {33, 37.2184}, {34, 37.2767}, {35, 37.343},
{36, 37.4225}, {37, 37.5307}, {38, 43.475}, {39, 43.5224}, {40, 43.5744}, {41, 43.6331}, {42, 43.7023}}

代码 4——每个 { } 里面有 3 个数。第 1 个数:  n = 0—97。第 2 个数:  每个 n 对应的最小解(四舍五入了)。第3个数:  找了个相近的数作比较。
代码 4——Table[{n, Round[x /. First@NSolve[{n == x*Cos[x], 6 + n > x > n}, {x}]], Round[2 Pi*Round[n/(2 Pi)]]}, {n, 0, 97}]
{{0, 2, 0}, {1, 5, 0}, {2, 5, 0}, {3, 5, 0}, {4, 6, 6}, {5, 6, 6}, {6, 6, 6}, {7, 12, 6}, {8, 12, 6}, {9, 12, 6}, {10, 12, 13}, {11, 12, 13}, {12, 12, 13}, {13, 18, 13}, {14, 18, 13}, {15, 18, 13}, {16, 18, 19}, {17, 18, 19},
{18, 19, 19}, {19, 24, 19}, {20, 25, 19}, {21, 25, 19}, {22, 25, 25}, {23, 25, 25}, {24, 25, 25}, {25, 25, 25}, {26, 31, 25}, {27, 31, 25}, {28, 31, 25}, {29, 31, 31}, {30, 31, 31}, {31, 31, 31}, {32, 37, 31}, {33, 37, 31},
{34, 37, 31}, {35, 37, 38}, {36, 37, 38}, {37, 38, 38}, {38, 43, 38}, {39, 44, 38}, {40, 44, 38}, {41, 44, 44}, {42, 44, 44}, {43, 44, 44}, {44, 50, 44}, {45, 50, 44}, {46, 50, 44}, {47, 50, 44}, {48, 50, 50}, {49, 50, 50},
{50, 50, 50}, {51, 56, 50}, {52, 56, 50}, {53, 56, 50}, {54, 56, 57}, {55, 56, 57}, {56, 56, 57}, {57, 62, 57}, {58, 62, 57}, {59, 62, 57}, {60, 63, 63}, {61, 63, 63}, {62, 63, 63}, {63, 69, 63}, {64, 69, 63}, {65, 69, 63},
{66, 69, 69}, {67, 69, 69}, {68, 69, 69}, {69, 69, 69}, {70, 75, 69}, {71, 75, 69}, {72, 75, 69}, {73, 75, 75}, {74, 75, 75}, {75, 75, 75}, {76, 81, 75}, {77, 81, 75}, {78, 81, 75}, {79, 81, 82}, {80, 81, 82}, {81, 82, 82},
{82, 88, 82}, {83, 88, 82}, {84, 88, 82}, {85, 88, 88}, {86, 88, 88}, {87, 88, 88}, {88, 94, 88}, {89, 94, 88}, {90, 94, 88}, {91, 94, 88}, {92, 94, 94}, {93, 94, 94}, {94, 94, 94}, {95, 100, 94}, {96, 100, 94}, {97, 100, 94},

代码 5——Table[Round[2 Pi*Round[n/(2 Pi)]], {n, 0, 169}]——把代码 4 中的第 3 个数单独提出来。
{0, 0, 0, 0, 6, 6, 6, 6, 6, 6, 13, 13, 13, 13, 13, 13, 19, 19, 19, 19, 19, 19, 25, 25, 25, 25, 25, 25, 25, 31, 31, 31, 31, 31, 31, 38, 38, 38, 38, 38, 38, 44, 44, 44, 44, 44, 44, 44, 50, 50, 50, 50, 50, 50, 57, 57, 57, 57, 57, 57,
63, 63, 63, 63, 63, 63, 69, 69, 69, 69, 69, 69, 69, 75, 75, 75, 75, 75, 75, 82, 82, 82, 82, 82, 82, 88, 88, 88, 88, 88, 88, 88, 94, 94, 94, 94, 94, 94, 101, 101, 101, 101, 101, 101, 107, 107, 107, 107, 107, 107, 113, 113,
113, 113, 113, 113, 113, 119, 119, 119, 119, 119, 119, 126, 126, 126, 126, 126, 126, 132, 132, 132, 132, 132, 132, 132, 138, 138, 138, 138, 138, 138, 145, 145, 145, 145, 145, 145, 151, 151, 151, 151, 151, 151, 157}

代码 6——Table[Round[2 Pi*n], {n, 0, 49}]——把代码 5 的重复数字去掉了。——你发现什么了吗？？？
{0, 6, 13, 19, 25, 31, 38, 44, 50, 57, 63, 69, 75, 82, 88, 94, 101, 107, 113, 119, 126, 132, 138, 145, 151, 157, 163, 170, 176, 182, 188, 195, 201, 207, 214, 220, 226, 232, 239, 245, 251, 258, 264, 270, 276, 283, 289, 295, 302, 308}