60.1.404 problem 406

Internal problem ID [10418]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 406
Date solved : Tuesday, January 28, 2025 at 04:49:39 PM
CAS classification : [_dAlembert]

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

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 269

dsolve(a*diff(y(x),x)^2-y(x)*diff(y(x),x)-x = 0,y(x), singsol=all)
 

\begin{align*} \frac {-\frac {\sqrt {2}\, \left (y+\sqrt {4 a x +y^{2}}\right ) \operatorname {arcsinh}\left (\frac {y+\sqrt {4 a x +y^{2}}}{2 a}\right )}{2}+x \sqrt {\frac {y \sqrt {4 a x +y^{2}}+2 a^{2}+2 a x +y^{2}}{a^{2}}}+c_{1} y+\sqrt {4 a x +y^{2}}\, c_{1}}{\sqrt {\frac {y \sqrt {4 a x +y^{2}}+y^{2}+2 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \frac {\frac {\sqrt {2}\, \left (y-\sqrt {4 a x +y^{2}}\right ) \operatorname {arcsinh}\left (\frac {-y+\sqrt {4 a x +y^{2}}}{2 a}\right )}{2}-\frac {c_{1} \sqrt {2}\, y}{2}+\frac {c_{1} \sqrt {2}\, \sqrt {4 a x +y^{2}}}{2}+x \sqrt {\frac {y^{2}-y \sqrt {4 a x +y^{2}}+2 a^{2}+2 a x}{a^{2}}}}{\sqrt {\frac {-y \sqrt {4 a x +y^{2}}+y^{2}+2 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.313 (sec). Leaf size: 117

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

\[ \text {Solve}\left [\left \{x=a \exp \left (\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right ) \int \frac {\exp \left (-\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right )}{K[1]+\frac {1}{K[1]}} \, dK[1]+c_1 \exp \left (\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right ),y(x)=a K[1]-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]