60.1.525 problem 528

Internal problem ID [10539]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 528
Date solved : Monday, January 27, 2025 at 08:52:30 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} {y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y+a b x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.057 (sec). Leaf size: 127

dsolve(diff(y(x),x)^3+a*diff(y(x),x)^2+b*y(x)+a*b*x=0,y(x), singsol=all)
 

\[ y = \frac {-{\mathrm e}^{3 \operatorname {RootOf}\left (-2 a^{2} \textit {\_Z} -3 \,{\mathrm e}^{2 \textit {\_Z}}+8 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -5 a^{2}-2 b x \right )}+2 \,{\mathrm e}^{2 \operatorname {RootOf}\left (-2 a^{2} \textit {\_Z} -3 \,{\mathrm e}^{2 \textit {\_Z}}+8 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -5 a^{2}-2 b x \right )} a -{\mathrm e}^{\operatorname {RootOf}\left (-2 a^{2} \textit {\_Z} -3 \,{\mathrm e}^{2 \textit {\_Z}}+8 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -5 a^{2}-2 b x \right )} a^{2}-a x b}{b} \]

✓ Solution by Mathematica

Time used: 0.146 (sec). Leaf size: 160

DSolve[a*b*x + b*y[x] + a*D[y[x],x]^2 + D[y[x],x]^3==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\left \{x=\int _1^{\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{-2 a^3-27 b x a-27 b y(x)+\sqrt {\left (-2 a^3-27 b x a-27 b y(x)\right )^2-4 a^6}}}-\frac {a}{3}+\frac {\sqrt [3]{-2 a^3-27 b x a-27 b y(x)+\sqrt {\left (-2 a^3-27 b x a-27 b y(x)\right )^2-4 a^6}}}{3 \sqrt [3]{2}}}\frac {-3 K[1]^2-2 a K[1]}{b (a+K[1])}dK[1]+c_1\right \},y(x)\right ] \]