20.15 problem 15

Internal problem ID [2352]

Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section: Exercise 38, page 173
Problem number: 15.
ODE order: 1.
ODE degree: 3.

CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]

\[ \boxed {\frac {1}{{y^{\prime }}^{2}}+y^{\prime } x -2 y=0} \]

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 2120

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

\begin{align*} \frac {\left (-442368 x y \left (x \right )^{5} \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}-\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {8}{3}}-16 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {7}{3}} y \left (x \right )-1024 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {5}{3}} y \left (x \right )^{3}-4864 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {4}{3}} y \left (x \right )^{4}-40960 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} y \left (x \right )^{6}-65536 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}} y \left (x \right )^{7}-1769472 y \left (x \right )^{8}+414720 x^{3} y \left (x \right )^{2} \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+6193152 x^{2} y \left (x \right )^{5}-3732480 x^{4} y \left (x \right )^{2}\right ) c_{1}}{{\left (\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+4 y \left (x \right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y \left (x \right )^{2}\right )}^{3} x \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}}+x -\frac {108 x^{3} \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )}{{\left (\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+4 y \left (x \right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y \left (x \right )^{2}\right )}^{3}} = 0 \frac {\left (-1769472 i \sqrt {3}\, y \left (x \right )^{8}+1024 i y \left (x \right )^{3} \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {5}{3}} \sqrt {3}+1244160 i \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}\, x^{3} y \left (x \right )^{2}-1327104 i \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}\, x y \left (x \right )^{5}+40960 i y \left (x \right )^{6} \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-3732480 i \sqrt {3}\, x^{4} y \left (x \right )^{2}+i \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {8}{3}} \sqrt {3}-442368 x y \left (x \right )^{5} \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}-\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {8}{3}}+32 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {7}{3}} y \left (x \right )-1024 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {5}{3}} y \left (x \right )^{3}+9728 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {4}{3}} y \left (x \right )^{4}-40960 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} y \left (x \right )^{6}+131072 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}} y \left (x \right )^{7}-1769472 y \left (x \right )^{8}+6193152 i \sqrt {3}\, x^{2} y \left (x \right )^{5}+414720 x^{3} y \left (x \right )^{2} \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+6193152 x^{2} y \left (x \right )^{5}-3732480 x^{4} y \left (x \right )^{2}\right ) c_{1}}{{\left (i \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 i \sqrt {3}\, y \left (x \right )^{2}-\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+8 y \left (x \right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 y \left (x \right )^{2}\right )}^{3} x \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}}+x -\frac {864 x^{3} \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )}{{\left (i \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 i \sqrt {3}\, y \left (x \right )^{2}-\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+8 y \left (x \right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 y \left (x \right )^{2}\right )}^{3}} = 0 \frac {\left (-1769472 i \sqrt {3}\, y \left (x \right )^{8}+1024 i y \left (x \right )^{3} \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {5}{3}} \sqrt {3}+1244160 i \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}\, x^{3} y \left (x \right )^{2}-1327104 i \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}\, x y \left (x \right )^{5}+40960 i y \left (x \right )^{6} \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-3732480 i \sqrt {3}\, x^{4} y \left (x \right )^{2}+i \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {8}{3}} \sqrt {3}+442368 x y \left (x \right )^{5} \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {8}{3}}-32 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {7}{3}} y \left (x \right )+1024 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {5}{3}} y \left (x \right )^{3}-9728 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {4}{3}} y \left (x \right )^{4}+40960 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} y \left (x \right )^{6}-131072 \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}} y \left (x \right )^{7}+1769472 y \left (x \right )^{8}+6193152 i \sqrt {3}\, x^{2} y \left (x \right )^{5}-414720 x^{3} y \left (x \right )^{2} \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}-6193152 x^{2} y \left (x \right )^{5}+3732480 x^{4} y \left (x \right )^{2}\right ) c_{1}}{{\left (i \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 i \sqrt {3}\, y \left (x \right )^{2}+\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (x \right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y \left (x \right )^{2}\right )}^{3} x \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}}+x +\frac {864 x^{3} \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )}{{\left (i \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 i \sqrt {3}\, y \left (x \right )^{2}+\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (x \right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y \left (x \right )^{2}\right )}^{3}} = 0 \end{align*}

✓ Solution by Mathematica

Time used: 149.881 (sec). Leaf size: 10773

DSolve[1/(y'[x]^2)+y'[x]*x==2*y[x],y[x],x,IncludeSingularSolutions -> True]
 

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