64.3.11 problem 12

Internal problem ID [13282]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.1 (Exact differential equations and integrating factors). Exercises page 37
Problem number : 12
Date solved : Tuesday, January 28, 2025 at 05:14:24 AM
CAS classification : [_exact, _rational]

\begin{align*} 3 x^{2} y^{2}-y^{3}+2 x +\left (2 x^{3} y-3 x y^{2}+1\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (-2\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 1.671 (sec). Leaf size: 208

dsolve([(3*x^2*y(x)^2-y(x)^3+2*x)+(2*x^3*y(x)-3*x*y(x)^2+1)*diff(y(x),x)=0,y(-2) = 1],y(x), singsol=all)
 

\[ y = \frac {-\frac {2^{{2}/{3}} \left (1+i \sqrt {3}\right ) {\left (\left (2 x^{7}+3 \sqrt {3}\, \sqrt {\frac {4 x^{10}+4 x^{8}+44 x^{5}+72 x^{3}+27 x -4}{x}}+36 x^{2}+27\right ) x^{2}\right )}^{{2}/{3}}}{2}+x \left (2 x^{2} {\left (\left (2 x^{7}+3 \sqrt {3}\, \sqrt {\frac {4 x^{10}+4 x^{8}+44 x^{5}+72 x^{3}+27 x -4}{x}}+36 x^{2}+27\right ) x^{2}\right )}^{{1}/{3}}+\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (x^{5}+3\right )\right )}{6 {\left (\left (2 x^{7}+3 \sqrt {3}\, \sqrt {\frac {4 x^{10}+4 x^{8}+44 x^{5}+72 x^{3}+27 x -4}{x}}+36 x^{2}+27\right ) x^{2}\right )}^{{1}/{3}} x} \]

✓ Solution by Mathematica

Time used: 60.242 (sec). Leaf size: 250

DSolve[{(3*x^2*y[x]^2-y[x]^3+2*x)+(2*x^3*y[x]-3*x*y[x]^2+1)*D[y[x],x]==0,{y[-2]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x^6+4 \sqrt [3]{-2 x^9-36 x^4-27 x^2+3 \sqrt {3} \sqrt {x^3 \left (4 x^{10}+4 x^8+44 x^5+72 x^3+27 x-4\right )}} x^3+\left (1+i \sqrt {3}\right ) \left (-4 x^9-72 x^4-54 x^2+6 \sqrt {3} \sqrt {x^3 \left (4 x^{10}+4 x^8+44 x^5+72 x^3+27 x-4\right )}\right )^{2/3}+6 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x}{12 x \sqrt [3]{-2 x^9-36 x^4-27 x^2+3 \sqrt {3} \sqrt {x^3 \left (4 x^{10}+4 x^8+44 x^5+72 x^3+27 x-4\right )}}} \]