60.1.45 problem 45

Internal problem ID [10059]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 45
Date solved : Tuesday, January 28, 2025 at 04:15:36 PM
CAS classification : [_Abel]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 113

dsolve(diff(y(x),x) + 2*(a^2*x^3 - b^2*x)*y(x)^3 + 3*b*y(x)^2=0,y(x), singsol=all)
 

\[ c_{1} +\frac {\left (\frac {a^{2} y^{2} x^{4}-y^{2} b^{2} x^{2}+2 b x y-1}{\left (b x y-1\right )^{2}}\right )^{{1}/{4}} a x}{\sqrt {\frac {a \,x^{2} y}{b x y-1}}\, b \left (b x y-1\right )}-\int _{}^{\frac {a \,x^{2} y}{b x y-1}}\frac {\left (\textit {\_a}^{2}-1\right )^{{1}/{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} = 0 \]

✓ Solution by Mathematica

Time used: 0.452 (sec). Leaf size: 133

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

\[ \text {Solve}\left [c_1=\sqrt [4]{\left (\frac {b}{a x}-\frac {1}{a x^2 y(x)}\right )^2-1} \left (-\frac {\left (\frac {b}{a x}-\frac {1}{a x^2 y(x)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\left (\frac {b}{a x}-\frac {1}{a x^2 y(x)}\right )^2\right )}{2 \sqrt [4]{1-\left (\frac {b}{a x}-\frac {1}{a x^2 y(x)}\right )^2}}-\frac {a x}{b}\right ),y(x)\right ] \]