15.20.15 problem 15

Internal problem ID [3323]
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
Date solved : Monday, January 27, 2025 at 07:33:40 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 1429

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

\begin{align*} \frac {279936 \left (\frac {x \left (\frac {4 x^{2} y \left (x \right )^{2}}{9}-\frac {16 y \left (x \right )^{3} c_{1}}{3}+c_{1} x^{2}\right ) \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}}{108}+\frac {40 y \left (x \right )^{3} c_{1} x^{2}}{81}-\frac {y \left (x \right )^{2} x^{4}}{27}-\frac {c_{1} x^{4}}{12}-\frac {32 y \left (x \right )^{6} c_{1}}{81}+\frac {8 x^{2} y \left (x \right )^{5}}{243}\right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{2}/{3}}+34992 \left (-\frac {16 x^{2} y \left (x \right )^{3}}{9}+\frac {64 c_{1} y \left (x \right )^{4}}{3}+x^{4}-\frac {32 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (-\frac {16 y \left (x \right )^{3}}{27}+x^{2}-\frac {x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}}{9}\right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{1}/{3}}+279936 y \left (x \right ) \left (-\frac {x \left (-\frac {32 x^{2} y \left (x \right )^{3}}{27}+\frac {128 c_{1} y \left (x \right )^{4}}{9}+x^{4}-\frac {40 x^{2} c_{1} y \left (x \right )}{3}\right ) \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}}{9}+\frac {128 y \left (x \right )^{6} x^{2}}{243}-\frac {16 x^{4} y \left (x \right )^{3}}{9}+x^{6}-\frac {512 y \left (x \right )^{7} c_{1}}{81}+\frac {1792 y \left (x \right )^{4} c_{1} x^{2}}{81}-\frac {40 y \left (x \right ) c_{1} x^{4}}{3}\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 )^{{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 )^{{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 )^{{1}/{3}}} &= 0 \\ \frac {1119744 \left (\frac {x \left (-\frac {16 x^{2} y \left (x \right )^{2}}{9}-\frac {16 y \left (x \right )^{3} c_{1}}{3}+c_{1} x^{2}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{144}-\frac {\left (i \sqrt {3}-1\right ) \left (\frac {128 x^{2} y \left (x \right )^{5}}{81}+\frac {128 y \left (x \right )^{6} c_{1}}{27}-\frac {16 y \left (x \right )^{2} x^{4}}{9}-\frac {160 y \left (x \right )^{3} c_{1} x^{2}}{27}+c_{1} x^{4}\right )}{48}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{2}/{3}}-279936 \left (-\frac {16 x^{2} y \left (x \right )^{3}}{9}-\frac {16 c_{1} y \left (x \right )^{4}}{3}+x^{4}+\frac {8 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (-\frac {x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{9}-\frac {16 y \left (x \right )^{3}}{27}+x^{2}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{1}/{3}}+1119744 y \left (x \right ) \left (-\frac {x \left (i+\frac {\sqrt {3}}{3}\right ) \left (-\frac {32 x^{2} y \left (x \right )^{3}}{27}-\frac {32 c_{1} y \left (x \right )^{4}}{9}+x^{4}+\frac {10 x^{2} c_{1} y \left (x \right )}{3}\right ) \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{3}+\left (\frac {128 y \left (x \right )^{6} x^{2}}{243}+\frac {128 y \left (x \right )^{7} c_{1}}{81}-\frac {16 x^{4} y \left (x \right )^{3}}{9}-\frac {448 y \left (x \right )^{4} c_{1} x^{2}}{81}+x^{6}+\frac {10 y \left (x \right ) c_{1} x^{4}}{3}\right ) \left (1+i \sqrt {3}\right )\right )}{\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{1}/{3}} \left (16 i \sqrt {3}\, y \left (x \right )^{2}-i \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{2}/{3}} \sqrt {3}+16 y \left (x \right )^{2}-8 y \left (x \right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{1}/{3}}+\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{2}/{3}}\right )^{3} x} &= 0 \\ \frac {1119744 \left (\frac {x \left (i+\frac {\sqrt {3}}{3}\right ) \left (-\frac {16 x^{2} y \left (x \right )^{2}}{9}-\frac {16 y \left (x \right )^{3} c_{1}}{3}+c_{1} x^{2}\right ) \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{144}-\frac {\left (1+i \sqrt {3}\right ) \left (\frac {128 x^{2} y \left (x \right )^{5}}{81}+\frac {128 y \left (x \right )^{6} c_{1}}{27}-\frac {16 y \left (x \right )^{2} x^{4}}{9}-\frac {160 y \left (x \right )^{3} c_{1} x^{2}}{27}+c_{1} x^{4}\right )}{48}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{2}/{3}}+279936 \left (-\frac {16 x^{2} y \left (x \right )^{3}}{9}-\frac {16 c_{1} y \left (x \right )^{4}}{3}+x^{4}+\frac {8 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (-\frac {x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{9}-\frac {16 y \left (x \right )^{3}}{27}+x^{2}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{1}/{3}}+1119744 y \left (x \right ) \left (-\frac {x \left (-\frac {32 x^{2} y \left (x \right )^{3}}{27}-\frac {32 c_{1} y \left (x \right )^{4}}{9}+x^{4}+\frac {10 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{3}+\left (\frac {128 y \left (x \right )^{6} x^{2}}{243}+\frac {128 y \left (x \right )^{7} c_{1}}{81}-\frac {16 x^{4} y \left (x \right )^{3}}{9}-\frac {448 y \left (x \right )^{4} c_{1} x^{2}}{81}+x^{6}+\frac {10 y \left (x \right ) c_{1} x^{4}}{3}\right ) \left (i \sqrt {3}-1\right )\right )}{\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{1}/{3}} \left (16 i \sqrt {3}\, y \left (x \right )^{2}-i \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{2}/{3}} \sqrt {3}-16 y \left (x \right )^{2}+8 y \left (x \right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{1}/{3}}-\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{{2}/{3}}\right )^{3} x} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 152.672 (sec). Leaf size: 10773

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

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