62.7.1 problem Ex 1

Internal problem ID [12824]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 14. Equations reducible to linear equations (Bernoulli). Page 21
Problem number : Ex 1
Date solved : Tuesday, January 28, 2025 at 04:24:55 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime }-2 \left (1+x \right ) y&=y^{{5}/{2}} \end{align*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 64

dsolve((1-x^2)*diff(y(x),x)-2*(1+x)*y(x)=y(x)^(5/2),y(x), singsol=all)
 

\[ -\frac {-1+\left (-\frac {3 \left (x -1\right )^{3} \ln \left (x -1\right )}{32}+\frac {3 \left (x -1\right )^{3} \ln \left (x +1\right )}{32}+c_{1} x^{3}+\left (-3 c_{1} -\frac {3}{16}\right ) x^{2}+\left (3 c_{1} +\frac {9}{16}\right ) x -c_{1} -\frac {5}{8}\right ) y^{{3}/{2}}}{y^{{3}/{2}}} = 0 \]

✓ Solution by Mathematica

Time used: 4.053 (sec). Leaf size: 98

DSolve[(1-x^2)*D[y[x],x]-2*(1+x)*y[x]==y[x]^(5/2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2^{2/3}}{\left ((x-1)^3 \left (-3 \int _1^x-\frac {1}{(K[1]-1)^4 (K[1]+1)}dK[1]+2 c_1\right )\right ){}^{2/3}} \\ y(x)\to 0 \\ y(x)\to \frac {\left (\frac {2}{3}\right )^{2/3}}{\left (-(x-1)^3 \int _1^x-\frac {1}{(K[1]-1)^4 (K[1]+1)}dK[1]\right ){}^{2/3}} \\ \end{align*}