Internal problem ID [8130]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 550.
ODE order: 1.
ODE degree: r.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{r}-a y^{s}-b \,x^{\frac {r s}{r -s}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.189 (sec). Leaf size: 64
dsolve(diff(y(x),x)^r-a*y(x)^s-b*x^(r*s/(r-s))=0,y(x), singsol=all)
\[ -\left (\int _{\textit {\_b}}^{y \relax (x )}\frac {1}{x \left (r -s \right ) \left (a \,\textit {\_a}^{s}+b \,x^{\frac {r s}{r -s}}\right )^{\frac {1}{r}}-r \textit {\_a}}d \textit {\_a} \right )+\frac {\ln \relax (x )}{r -s}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.71 (sec). Leaf size: 488
DSolve[-(b*x^((r*s)/(r - s))) - a*y[x]^s + y'[x]^r==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {r}{-r x \left (a K[2]^s+b x^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}+s x \left (a K[2]^s+b x^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}+r K[2]}-\int _1^x\left (\frac {a s K[2]^{s-1} \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}-1}}{r K[1] \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-s K[1] \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-r K[2]}-\frac {r \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}} \left (-\frac {a s^2 K[1] K[2]^{s-1} \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}-1}}{r}+a s K[1] K[2]^{s-1} \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}-1}-r\right )}{\left (r K[1] \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-s K[1] \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-r K[2]\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {r \left (a y(x)^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}}{r K[1] \left (a y(x)^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-s K[1] \left (a y(x)^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-r y(x)}dK[1]=c_1,y(x)\right ] \]