38.2.27 problem 27

Internal problem ID [6456]
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
Date solved : Monday, January 27, 2025 at 02:04:38 PM
CAS classification : [_separable]

\begin{align*} x y y^{\prime }-\left (1+x \right ) \sqrt {y-1}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 22

dsolve(x*y(x)*diff(y(x),x)-(1+x)*sqrt(y(x)-1)=0,y(x), singsol=all)
 

\[ \frac {\left (-2 y-4\right ) \sqrt {y-1}}{3}+x +c_1 +\ln \left (x \right ) = 0 \]

✓ Solution by Mathematica

Time used: 4.628 (sec). Leaf size: 582

DSolve[x*y[x]*D[y[x],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*}