Internal problem ID [8719]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 383.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`]]
\[ \boxed {{y^{\prime }}^{2}+a x y^{\prime }+b y=-c \,x^{2}} \]
✗ Solution by Maple
dsolve(diff(y(x),x)^2+a*x*diff(y(x),x)+b*y(x)+c*x^2 = 0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 2.085 (sec). Leaf size: 1085
DSolve[c*x^2 + b*y[x] + a*x*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\text {RootSum}\left [\text {$\#$1}^4-2 \text {$\#$1}^3 b-2 \text {$\#$1}^2 a^2-4 \text {$\#$1}^2 a b+8 \text {$\#$1}^2 c-2 \text {$\#$1} a^2 b+8 \text {$\#$1} b c+a^4-8 a^2 c+16 c^2\&,\frac {-\text {$\#$1}^3 \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )+\text {$\#$1}^3 \log (x)+\text {$\#$1}^2 b \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )-\text {$\#$1}^2 b \log (x)+\text {$\#$1} a^2 \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )+2 \text {$\#$1} a b \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )-4 \text {$\#$1} c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )+a^2 b \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )-4 b c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )-\text {$\#$1} a^2 \log (x)-2 \text {$\#$1} a b \log (x)+4 \text {$\#$1} c \log (x)-a^2 b \log (x)+4 b c \log (x)}{-2 \text {$\#$1}^3+3 \text {$\#$1}^2 b+2 \text {$\#$1} a^2+4 \text {$\#$1} a b-8 \text {$\#$1} c+a^2 b-4 b c}\&\right ]-\log \left (\sqrt {-b y(x)} \sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 b y(x)\right )+\frac {1}{2} \log (y(x))+2 \log (x)&=c_1,y(x)\right ] \\ \text {Solve}\left [\text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^3 b-2 \text {$\#$1}^2 a^2-4 \text {$\#$1}^2 a b+8 \text {$\#$1}^2 c+2 \text {$\#$1} a^2 b-8 \text {$\#$1} b c+a^4-8 a^2 c+16 c^2\&,\frac {\text {$\#$1}^3 \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )+\text {$\#$1}^3 (-\log (x))+\text {$\#$1}^2 b \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )-\text {$\#$1}^2 b \log (x)-\text {$\#$1} a^2 \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )-2 \text {$\#$1} a b \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )+4 \text {$\#$1} c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )+a^2 b \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )-4 b c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 \sqrt {-b y(x)}\right )+\text {$\#$1} a^2 \log (x)+2 \text {$\#$1} a b \log (x)-4 \text {$\#$1} c \log (x)-a^2 b \log (x)+4 b c \log (x)}{2 \text {$\#$1}^3+3 \text {$\#$1}^2 b-2 \text {$\#$1} a^2-4 \text {$\#$1} a b+8 \text {$\#$1} c+a^2 b-4 b c}\&\right ]-\log \left (\sqrt {-b y(x)} \sqrt {x^2 \left (a^2-4 c\right )-4 b y(x)}+2 b y(x)\right )+\frac {1}{2} \log (y(x))+2 \log (x)&=c_1,y(x)\right ] \\ \end{align*}