60.4.73 problem 1523

Internal problem ID [11524]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1523
Date solved : Tuesday, January 28, 2025 at 06:06:45 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.126 (sec). Leaf size: 150

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

\[ y(x)\to \frac {1}{2} x \left (-2 c_3 \left (x^2-3 x+1\right ) \int _1^x\frac {\left (\int \exp \left (\int _1^{K[2]}\frac {-K[1]^6+6 K[1]^5-9 K[1]^4+18 K[1]^3-7 K[1]^2+1}{K[1] \left (K[1]^6-3 K[1]^5+K[1]^4-K[1]^2+3 K[1]-1\right )}dK[1]\right ) \, dK[2]\right ) \left (K[2]^2-1\right )}{2 \left (K[2]^2-3 K[2]+1\right )^2}dK[2]-2 c_1 \left (x^2-3 x+1\right )+c_2 (x-1)^2\right ) \]