56.1.18 problem 18

Internal problem ID [8730]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 18
Date solved : Monday, January 27, 2025 at 04:20:52 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\sqrt {\frac {y+1}{y^{2}}} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.442 (sec). Leaf size: 146

dsolve([diff(y(x),x)=sqrt( (1+y(x))/y(x)^2),y(0) = 1],y(x), singsol=all)
 

\[ y = -\frac {\left (1+i \sqrt {3}\right ) \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{{1}/{3}}+4}{4 \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{{1}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.078 (sec). Leaf size: 123

DSolve[{D[y[x],x]==Sqrt[ (1+y[x])/y[x]^2],y[0]==1},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^2+\sqrt {81 x^4-216 \sqrt {2} x^3+288 x^2-64}-12 \sqrt {2} x}+\frac {i \left (\sqrt {3}+i\right )}{\sqrt [3]{9 x^2+\sqrt {81 x^4-216 \sqrt {2} x^3+288 x^2-64}-12 \sqrt {2} x}}+1 \]