60.3.390 problem 1396

Internal problem ID [11400]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1396
Date solved : Monday, January 27, 2025 at 11:19:23 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }&=-\frac {A y}{\left (a \,x^{2}+b x +c \right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 174

dsolve(diff(diff(y(x),x),x) = -A/(a*x^2+b*x+c)^2*y(x),y(x), singsol=all)
 

\[ y = \left ({\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{2} +{\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{1} \right ) \sqrt {a \,x^{2}+b x +c} \]

✓ Solution by Mathematica

Time used: 0.409 (sec). Leaf size: 154

DSolve[D[y[x],{x,2}] == -((A*y[x])/(c + b*x + a*x^2)^2),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right )dK[2]+c_1\right ) \]