57.1.59 problem 59

Internal problem ID [9043]
Book : First order enumerated odes
Section : section 1
Problem number : 59
Date solved : Monday, January 27, 2025 at 05:27:56 PM
CAS classification : [_separable]

\begin{align*} {y^{\prime }}^{2}&=\frac {1}{x^{3} y^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.051 (sec). Leaf size: 133

dsolve(diff(y(x),x)^2=1/(x^3*y(x)^4),y(x), singsol=all)
 

\begin{align*} y &= \left (\frac {\sqrt {x}\, c_{1} -6}{\sqrt {x}}\right )^{{1}/{3}} \\ y &= -\frac {\left (\frac {\sqrt {x}\, c_{1} -6}{\sqrt {x}}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left (\frac {\sqrt {x}\, c_{1} -6}{\sqrt {x}}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\ y &= \left (\frac {\sqrt {x}\, c_{1} +6}{\sqrt {x}}\right )^{{1}/{3}} \\ y &= -\frac {\left (\frac {\sqrt {x}\, c_{1} +6}{\sqrt {x}}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left (\frac {\sqrt {x}\, c_{1} +6}{\sqrt {x}}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.402 (sec). Leaf size: 157

DSolve[(D[y[x],x])^2==1/(x^3*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt [3]{-3} \sqrt [3]{-\frac {2}{\sqrt {x}}+c_1} \\ y(x)\to \sqrt [3]{3} \sqrt [3]{-\frac {2}{\sqrt {x}}+c_1} \\ y(x)\to (-1)^{2/3} \sqrt [3]{3} \sqrt [3]{-\frac {2}{\sqrt {x}}+c_1} \\ y(x)\to -\sqrt [3]{-3} \sqrt [3]{\frac {2}{\sqrt {x}}+c_1} \\ y(x)\to \sqrt [3]{3} \sqrt [3]{\frac {2}{\sqrt {x}}+c_1} \\ y(x)\to (-1)^{2/3} \sqrt [3]{3} \sqrt [3]{\frac {2}{\sqrt {x}}+c_1} \\ \end{align*}