29.30.33 problem 893

Internal problem ID [5473]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 30
Problem number : 893
Date solved : Monday, January 27, 2025 at 11:25:17 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x^{2} {y^{\prime }}^{2}+x^{2}-y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 44

dsolve(x^2*diff(y(x),x)^2+x^2-y(x)^2 = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {x \left (\operatorname {LambertW}\left (-{\mathrm e} c_{1} x^{4}\right )-1\right )}{2 \operatorname {LambertW}\left (-{\mathrm e} c_{1} x^{4}\right ) \sqrt {-\frac {1}{\operatorname {LambertW}\left (-{\mathrm e} c_{1} x^{4}\right )}}} \]

✓ Solution by Mathematica

Time used: 7.539 (sec). Leaf size: 183

DSolve[x^2 (D[y[x],x])^2+x^2-y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+1}}{\sqrt {\frac {y(x)}{x}-1}}\right )-\frac {\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+1}}{\left (\sqrt {\frac {y(x)}{x}-1}-\sqrt {\frac {y(x)}{x}+1}\right )^2}&=\log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+1}}{\sqrt {\frac {y(x)}{x}-1}}\right )-\frac {\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+1}}{\left (\sqrt {\frac {y(x)}{x}-1}+\sqrt {\frac {y(x)}{x}+1}\right )^2}&=-\log (x)+c_1,y(x)\right ] \\ \end{align*}