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3

The equation is separable $$\frac{dt}{dh}=h^{-j}$$ So $$t+c_1=\frac{h^{1-j}}{1-j}$$ Solving for $h$ gives in the most general case $$h=\Big((j-1) (-c_1-t)\Big)^{\frac{1}{1-j}}=\Big((j-1) (c_2-t)\Big)^{\frac{1}{1-j}}$$ which is what Wolfram Alpha returns.

1

You can simplify your life (and WA's too) taking into account the fact that $x,y,a,b$ are constants for the problem. So, defining some intermediate numbers, your equations write $$A=s+p$$ $$B=q+t$$ $$C=\sqrt{p^2+q^2}+\sqrt{s^2+t^2}$$ $$D=p^2+q^2+s^2+t^2$$ and the result is just a monster ! You can eliminate variables $s$ and $t$ using the first equations ...

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Type Solve [{ x+a==s+p && y+b==q+t && Sqrt[x^2+y^2]+Sqrt[a^2+b^2]==Sqrt[s^2+t^2]+Sqrt[p^2+q^2] && x^2+y^2+a^2+b^2=(4/5)*(s^2+t^2+p^2+q^2) }, {s, t, p, q}] Then wolfram mathematica works correctly. Evaluation takes a few minutes... The answer is too long to display. It can print the answer and this takes also a few ...

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The reason that $\sqrt[250]{(-1)^{114}}$ seems to only be $1$ is that you're forgetting something: $\sqrt[n]{1}$ is always really $\pm1$, meaning that $-1$ is also a valid solution. The reason that $(\sqrt[250]{-1})^{114}$ gives you a complex number is that the calculator is following the order of operations and finding the $250$-root of $-1$ (which is $i$) ...

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When people set variables for programming, the variables are giving declarations of more than a letter many times so it is discriptive. For this reason, wolfram will read a concatenation of variables as one variable not two unless specified by parenthesis or multiplication. If we had a times b times c, we couldn't enter abc since that will be treated as one ...

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The following all work: ReplaceAll[Diff[Sin[x], x] , {x -> 0}] D[Sin[x], x] /. {x -> 0} Diff[Sin[x], x] where x = 0 Sin'(0)

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