Internal problem ID [5113]
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]
\[ \boxed {x y y^{\prime }-\left (x +1\right ) \sqrt {y-1}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 22
dsolve(x*y(x)*diff(y(x),x)-(1+x)*sqrt(y(x)-1)=0,y(x), singsol=all)
\[ \frac {\left (-2 y \left (x \right )-4\right ) \sqrt {y \left (x \right )-1}}{3}+x +c_{1} +\ln \left (x \right ) = 0 \]
✓ Solution by Mathematica
Time used: 5.614 (sec). Leaf size: 582
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^2+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 x^2+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+16+9 c_1{}^2\right )}+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+8+9 c_1{}^2}+\frac {2}{\sqrt [3]{9 x^2+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 x^2+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+16+9 c_1{}^2\right )}+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+8+9 c_1{}^2}}-1 \\ y(x)\to \frac {1}{4} i \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^2+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 x^2+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+16+9 c_1{}^2\right )}+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+8+9 c_1{}^2}+\frac {-1-i \sqrt {3}}{\sqrt [3]{9 x^2+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 x^2+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+16+9 c_1{}^2\right )}+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+8+9 c_1{}^2}}-1 \\ y(x)\to -\frac {1}{4} i \left (\sqrt {3}-i\right ) \sqrt [3]{9 x^2+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 x^2+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+16+9 c_1{}^2\right )}+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+8+9 c_1{}^2}+\frac {-1+i \sqrt {3}}{\sqrt [3]{9 x^2+3 \sqrt {(x+\log (x)+c_1){}^2 \left (9 x^2+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+16+9 c_1{}^2\right )}+9 \log ^2(x)+18 c_1 x+18 (x+c_1) \log (x)+8+9 c_1{}^2}}-1 \\ y(x)\to 1 \\ \end{align*}