Internal problem ID [11043]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 209.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{3} a +b \,x^{2}+x c +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-x \lambda +x^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 85
dsolve((a*x^3+b*x^2+c*x+d)*diff(y(x),x$2)+(x^3+lambda^3)*diff(y(x),x)-(x^2-lambda*x+lambda^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x +\lambda \right )+c_{2} \left (x +\lambda \right ) \left (\int {\mathrm e}^{\int \frac {-x^{4}+\left (-2 a -\lambda \right ) x^{3}-2 b \,x^{2}+\left (-\lambda ^{3}-2 c \right ) x -\lambda ^{4}-2 d}{\left (a \,x^{3}+b \,x^{2}+c x +d \right ) \left (x +\lambda \right )}d x}d x \right ) \]
✓ Solution by Mathematica
Time used: 1.343 (sec). Leaf size: 240
DSolve[(a*x^3+b*x^2+c*x+d)*y''[x]+(x^3+\[Lambda]^3)*y'[x]-(x^2-\[Lambda]*x+\[Lambda]^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_2 (\lambda +x) \int _1^x\exp \left (-\frac {\lambda +K[1]+2 a \log (\lambda +K[1])+\text {RootSum}\left [-a \lambda ^3+b \lambda ^2+3 a \text {$\#$1} \lambda ^2-3 a \text {$\#$1}^2 \lambda -c \lambda -2 b \text {$\#$1} \lambda +a \text {$\#$1}^3+b \text {$\#$1}^2+d+c \text {$\#$1}\&,\frac {a \log (\lambda +K[1]-\text {$\#$1}) \lambda ^3-b \log (\lambda +K[1]-\text {$\#$1}) \lambda ^2+c \log (\lambda +K[1]-\text {$\#$1}) \lambda +2 b \log (\lambda +K[1]-\text {$\#$1}) \text {$\#$1} \lambda -b \log (\lambda +K[1]-\text {$\#$1}) \text {$\#$1}^2-d \log (\lambda +K[1]-\text {$\#$1})-c \log (\lambda +K[1]-\text {$\#$1}) \text {$\#$1}}{3 a \lambda ^2-2 b \lambda -6 a \text {$\#$1} \lambda +3 a \text {$\#$1}^2+c+2 b \text {$\#$1}}\&\right ]}{a}\right )dK[1]}{\lambda }+\frac {c_1 (\lambda +x)}{\lambda } \]