60.1.65 problem 65

Internal problem ID [10079]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 65
Date solved : Monday, January 27, 2025 at 06:24:05 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }-\sqrt {\frac {y^{3}+1}{x^{3}+1}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 47

dsolve(diff(y(x),x) - sqrt((y(x)^3+1)/(x^3+1))=0,y(x), singsol=all)
 

\[ \int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{3}+1}}d \textit {\_a} -\frac {\int _{}^{x}\sqrt {\frac {y^{3}+1}{\textit {\_a}^{3}+1}}d \textit {\_a}}{\sqrt {y^{3}+1}}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 96.501 (sec). Leaf size: 337

DSolve[D[y[x],x] - Sqrt[(y[x]^3+1)/(x^3+1)]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {i (\text {$\#$1}+1) \sqrt {1+\frac {6 i}{\left (\sqrt {3}-3 i\right ) (\text {$\#$1}+1)}} \sqrt {\frac {2}{3}-\frac {4 i}{\left (\sqrt {3}+3 i\right ) (\text {$\#$1}+1)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {\text {$\#$1}+1}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {\text {$\#$1}^2-\text {$\#$1}+1}}\&\right ]\left [\frac {i (x+1) \sqrt {1+\frac {6 i}{\left (\sqrt {3}-3 i\right ) (x+1)}} \sqrt {\frac {2}{3}-\frac {4 i}{\left (\sqrt {3}+3 i\right ) (x+1)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {x+1}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {x^2-x+1}}+c_1\right ] \\ y(x)\to -1 \\ y(x)\to \sqrt [3]{-1} \\ y(x)\to -(-1)^{2/3} \\ \end{align*}