60.1.412 problem 414

Internal problem ID [10426]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 414
Date solved : Monday, January 27, 2025 at 07:43:10 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.087 (sec). Leaf size: 272

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

\begin{align*} -\int _{\textit {\_b}}^{x}\frac {y-\sqrt {-4 \textit {\_a}^{4}+y^{2}}}{\textit {\_a} \left (5 y-\sqrt {-4 \textit {\_a}^{4}+y^{2}}\right )}d \textit {\_a} -2 \left (\int _{}^{y}\frac {1+\left (40 \textit {\_f} -8 \sqrt {-4 x^{4}+\textit {\_f}^{2}}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (-5 \textit {\_f} +\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}\right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{5 \textit {\_f} -\sqrt {-4 x^{4}+\textit {\_f}^{2}}}d \textit {\_f} \right )+c_{1} &= 0 \\ -\int _{\textit {\_b}}^{x}\frac {y+\sqrt {-4 \textit {\_a}^{4}+y^{2}}}{\left (\sqrt {-4 \textit {\_a}^{4}+y^{2}}+5 y\right ) \textit {\_a}}d \textit {\_a} +2 \left (\int _{}^{y}\frac {-1+8 \left (\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f}}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.092 (sec). Leaf size: 107

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

\begin{align*} y(x)\to x^2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{5 K[2]+\sqrt {K[2]^2-4}}dK[2]\&\right ]\left [\int _1^x-\frac {1}{2 K[3]}dK[3]+c_1\right ] \\ y(x)\to x^2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {K[4]^2-4}-5 K[4]}dK[4]\&\right ]\left [\int _1^x\frac {1}{2 K[5]}dK[5]+c_1\right ] \\ \end{align*}