60.4.71 problem 1521

Internal problem ID [11522]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1521
Date solved : Tuesday, January 28, 2025 at 06:06:44 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} \left (1+x \right ) x^{3} y^{\prime \prime \prime }-\left (4 x +2\right ) x^{2} y^{\prime \prime }+\left (10 x +4\right ) x y^{\prime }-4 \left (1+3 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 28

dsolve((x+1)*x^3*diff(diff(diff(y(x),x),x),x)-(4*x+2)*x^2*diff(diff(y(x),x),x)+(10*x+4)*x*diff(y(x),x)-4*(3*x+1)*y(x)=0,y(x), singsol=all)
 

\[ y = x \left (\ln \left (x \right )^{2} c_3 x +c_{2} \ln \left (x \right ) x +x^{2} c_3 +c_{1} x +c_3 \right ) \]

✓ Solution by Mathematica

Time used: 0.109 (sec). Leaf size: 158

DSolve[-4*(1 + 3*x)*y[x] + x*(4 + 10*x)*D[y[x],x] - x^2*(2 + 4*x)*D[y[x],{x,2}] + x^3*(1 + x)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x^2 \left (c_2 \int _1^x\exp \left (\int _1^{K[4]}\frac {1}{K[1]^2+K[1]}dK[1]-\frac {1}{2} \int _1^{K[4]}\frac {2 K[2]+4}{K[2]^2+K[2]}dK[2]\right )dK[4]+c_3 \int _1^x\exp \left (\int _1^{K[5]}\frac {1}{K[1]^2+K[1]}dK[1]-\frac {1}{2} \int _1^{K[5]}\frac {2 K[2]+4}{K[2]^2+K[2]}dK[2]\right ) \int _1^{K[5]}\exp \left (-2 \int _1^{K[3]}\frac {1}{K[1]^2+K[1]}dK[1]\right )dK[3]dK[5]+c_1\right ) \]