40.4.22 problem 23 (b)

Internal problem ID [6662]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 6. Equations of first order and first degree (Linear equations). Supplemetary problems. Page 39
Problem number : 23 (b)
Date solved : Tuesday, January 28, 2025 at 03:10:30 PM
CAS classification : [_rational]

\begin{align*} 4 x^{2} y y^{\prime }&=3 x \left (3 y^{2}+2\right )+2 \left (3 y^{2}+2\right )^{3} \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 175

dsolve(4*x^2*y(x)*diff(y(x),x)=3*x*(3*y(x)^2+2)+2*(3*y(x)^2+2)^3,y(x), singsol=all)
 

\begin{align*} y &= -\frac {\sqrt {6}\, \sqrt {\frac {-3 c_1 \,x^{8}+\sqrt {-3 \left (c_1 \,x^{8}-\frac {1}{3}\right ) c_1 \,x^{9}}+1}{3 c_1 \,x^{8}-1}}}{3} \\ y &= \frac {\sqrt {6}\, \sqrt {\frac {-3 c_1 \,x^{8}+\sqrt {-3 \left (c_1 \,x^{8}-\frac {1}{3}\right ) c_1 \,x^{9}}+1}{3 c_1 \,x^{8}-1}}}{3} \\ y &= -\frac {\sqrt {\frac {-18 c_1 \,x^{8}-6 \sqrt {-3 \left (c_1 \,x^{8}-\frac {1}{3}\right ) c_1 \,x^{9}}+6}{3 c_1 \,x^{8}-1}}}{3} \\ y &= \frac {\sqrt {\frac {-18 c_1 \,x^{8}-6 \sqrt {-3 \left (c_1 \,x^{8}-\frac {1}{3}\right ) c_1 \,x^{9}}+6}{3 c_1 \,x^{8}-1}}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 22.093 (sec). Leaf size: 277

DSolve[4*x^2*y[x]*D[y[x],x]==3*x*(3*y[x]^2+2)+2*(3*y[x]^2+2)^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{3} \sqrt {2} \sqrt {-\frac {3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}+216 c_1}{x^8+72 c_1}} \\ y(x)\to \frac {1}{3} \sqrt {2} \sqrt {-\frac {3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}+216 c_1}{x^8+72 c_1}} \\ y(x)\to -\frac {1}{3} \sqrt {2} \sqrt {\frac {-3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}-216 c_1}{x^8+72 c_1}} \\ y(x)\to \frac {1}{3} \sqrt {2} \sqrt {\frac {-3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}-216 c_1}{x^8+72 c_1}} \\ y(x)\to -i \sqrt {\frac {2}{3}} \\ y(x)\to i \sqrt {\frac {2}{3}} \\ \end{align*}