Internal problem ID [4605]
Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY.
2001
Section: Program 24. First order differential equations. Further problems 24. page 1068
Problem number: 27.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {x y y^{\prime }-\left (1+x \right ) \sqrt {y-1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 20
dsolve(x*y(x)*diff(y(x),x)-(1+x)*sqrt(y(x)-1)=0,y(x), singsol=all)
\[ x +\ln \relax (x )-\frac {2 \sqrt {y \relax (x )-1}\, \left (y \relax (x )+2\right )}{3}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.434 (sec). Leaf size: 475
DSolve[x*y[x]*y'[x]-(1+x)*Sqrt[y[x]-1]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \sqrt [3]{9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+16\right )}+8}+\frac {2}{\sqrt [3]{9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+16\right )}+8}}-1 \\ y(x)\to \frac {1}{4} i \left (\sqrt {3}+i\right ) \sqrt [3]{9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+16\right )}+8}+\frac {-1-i \sqrt {3}}{\sqrt [3]{9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+16\right )}+8}}-1 \\ y(x)\to -\frac {1}{4} i \left (\sqrt {3}-i\right ) \sqrt [3]{9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+16\right )}+8}+\frac {i \left (\sqrt {3}+i\right )}{\sqrt [3]{9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 (x+c_1){}^2+9 \log (x) (\log (x)+2 (x+c_1))+16\right )}+8}}-1 \\ y(x)\to 1 \\ \end{align*}