60.3.315 problem 1321

Internal problem ID [11325]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1321
Date solved : Monday, January 27, 2025 at 11:13:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 15

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

\[ y = x \left (c_{2} \ln \left (x \right )+c_{2} x +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.177 (sec). Leaf size: 95

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

\[ y(x)\to \exp \left (\int _1^x\frac {1}{2 K[1]^2+2 K[1]}dK[1]-\frac {1}{2} \int _1^x-\frac {2 K[2]+1}{K[2]^2+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {1}{2 K[1]^2+2 K[1]}dK[1]\right )dK[3]+c_1\right ) \]