15.7.14 problem 14

Internal problem ID [2995]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 11, page 45
Problem number : 14
Date solved : Monday, January 27, 2025 at 07:06:14 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 3 y^{\prime }+\frac {2 y}{1+x}&=\frac {x}{y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 135

dsolve(3*diff(y(x),x)+2*y(x)/(x+1)=x/y(x)^2,y(x), singsol=all)
 

\begin{align*} y &= \frac {{\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+12 c_{1} \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}} 18^{{1}/{3}}}{6 \left (x +1\right )^{2}} \\ y &= -\frac {18^{{1}/{3}} {\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+12 c_{1} \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{12 \left (x +1\right )^{2}} \\ y &= \frac {18^{{1}/{3}} {\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+12 c_{1} \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{12 \left (x +1\right )^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.283 (sec). Leaf size: 144

DSolve[3*D[y[x],x]+2*y[x]/(x+1)==x/y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{\frac {3 x^4+8 x^3+6 x^2+12 c_1}{(x+1)^2}}}{2^{2/3} \sqrt [3]{3}} \\ y(x)\to -\frac {\sqrt [3]{-\frac {1}{3}} \sqrt [3]{\frac {3 x^4+8 x^3+6 x^2+12 c_1}{(x+1)^2}}}{2^{2/3}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{\frac {3 x^4+8 x^3+6 x^2+12 c_1}{(x+1)^2}}}{2^{2/3} \sqrt [3]{3}} \\ \end{align*}