15.20.11 problem 11

Internal problem ID [3319]
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 : 11
Date solved : Monday, January 27, 2025 at 07:33:26 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} 2 x {y^{\prime }}^{3}+1&=y {y^{\prime }}^{2} \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 505

dsolve(2*diff(y(x),x)^3*x+1=diff(y(x),x)^2*y(x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {9 \,3^{{1}/{3}} x^{2} \left (-\frac {2 \,3^{{2}/{3}} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}} c_{1}^{2}}{9}+\left (-\frac {2 \,3^{{1}/{3}} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}} c_{1}}{9}+x \right ) x \left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right )\right )}{{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}} \left (c_{1} 3^{{1}/{3}} x +{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}}\right )^{2}} \\ y \left (x \right ) &= \frac {4 \left (3 \left (-i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right ) c_{1}^{2} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}}+x \left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) \left (\left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_{1} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}}+9 x \right )\right ) x^{2} 3^{{1}/{3}}}{{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}} {\left ({\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}} \left (i-\sqrt {3}\right )+x c_{1} \left (i 3^{{1}/{3}}+3^{{5}/{6}}\right )\right )}^{2}} \\ y \left (x \right ) &= -\frac {4 x^{2} 3^{{1}/{3}} \left (-3 c_{1}^{2} \left (i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}}+x \left (c_{1} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}}-9 x \right ) \left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right )\right )}{{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}} {\left (\left (\sqrt {3}+i\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}}+x \left (-3^{{5}/{6}}+i 3^{{1}/{3}}\right ) c_{1} \right )}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 152.912 (sec). Leaf size: 17695

DSolve[2*D[y[x],x]^3*x+1==D[y[x],x]^2*y[x],y[x],x,IncludeSingularSolutions -> True]
 

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