60.1.382 problem 383

Internal problem ID [10396]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 383
Date solved : Monday, January 27, 2025 at 07:38:58 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} {y^{\prime }}^{2}+a x y^{\prime }+b y+c \,x^{2}&=0 \end{align*}

✗ 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: 10.174 (sec). Leaf size: 1118

DSolve[c*x^2 + b*y[x] + a*x*D[y[x],x] + D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\int _1^{y(x)}\left (-\frac {\sqrt {a^2 x^2-4 c x^2-4 b K[2]} x}{2 \left (c x^4+2 a K[2] x^2+b K[2] x^2+4 K[2]^2\right )}-\int _1^x\left (\frac {a K[1]+b K[1]}{c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2}+\frac {\sqrt {a^2 K[1]^2-4 c K[1]^2-4 b K[2]}}{c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2}-\frac {2 b K[2]}{\sqrt {a^2 K[1]^2-4 c K[1]^2-4 b K[2]} \left (c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2\right )}-\frac {\left (2 a K[1]^2+b K[1]^2+8 K[2]\right ) \left (c K[1]^3+a K[2] K[1]+b K[2] K[1]\right )}{\left (c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2\right )^2}-\frac {K[2] \left (2 a K[1]^2+b K[1]^2+8 K[2]\right ) \sqrt {a^2 K[1]^2-4 c K[1]^2-4 b K[2]}}{\left (c K[1]^4+2 a K[2] K[1]^2+b K[2] K[1]^2+4 K[2]^2\right )^2}\right )dK[1]+\frac {a x^2+4 K[2]}{2 \left (c x^4+2 a K[2] x^2+b K[2] x^2+4 K[2]^2\right )}\right )dK[2]+\int _1^x\left (\frac {\sqrt {a^2 K[1]^2-4 c K[1]^2-4 b y(x)} y(x)}{c K[1]^4+2 a y(x) K[1]^2+b y(x) K[1]^2+4 y(x)^2}+\frac {c K[1]^3+a y(x) K[1]+b y(x) K[1]}{c K[1]^4+2 a y(x) K[1]^2+b y(x) K[1]^2+4 y(x)^2}\right )dK[1]&=c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {a^2 x^2-4 c x^2-4 b K[4]} x}{2 \left (c x^4+2 a K[4] x^2+b K[4] x^2+4 K[4]^2\right )}-\int _1^x\left (\frac {a K[3]+b K[3]}{c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2}-\frac {\sqrt {a^2 K[3]^2-4 c K[3]^2-4 b K[4]}}{c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2}+\frac {2 b K[4]}{\sqrt {a^2 K[3]^2-4 c K[3]^2-4 b K[4]} \left (c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2\right )}-\frac {\left (2 a K[3]^2+b K[3]^2+8 K[4]\right ) \left (c K[3]^3+a K[4] K[3]+b K[4] K[3]\right )}{\left (c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2\right )^2}+\frac {K[4] \left (2 a K[3]^2+b K[3]^2+8 K[4]\right ) \sqrt {a^2 K[3]^2-4 c K[3]^2-4 b K[4]}}{\left (c K[3]^4+2 a K[4] K[3]^2+b K[4] K[3]^2+4 K[4]^2\right )^2}\right )dK[3]+\frac {a x^2+4 K[4]}{2 \left (c x^4+2 a K[4] x^2+b K[4] x^2+4 K[4]^2\right )}\right )dK[4]+\int _1^x\left (\frac {c K[3]^3+a y(x) K[3]+b y(x) K[3]}{c K[3]^4+2 a y(x) K[3]^2+b y(x) K[3]^2+4 y(x)^2}-\frac {y(x) \sqrt {a^2 K[3]^2-4 c K[3]^2-4 b y(x)}}{c K[3]^4+2 a y(x) K[3]^2+b y(x) K[3]^2+4 y(x)^2}\right )dK[3]&=c_1,y(x)\right ] \\ \end{align*}