23.2.7 problem 7

Internal problem ID [5362]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 7
Date solved : Tuesday, September 30, 2025 at 12:36:11 PM
CAS classification : [[_homogeneous, `class G`]]

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

✗ Maple

ode:=diff(y(x),x)^2+x^2*a+b*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.878 (sec). Leaf size: 581

ode=(D[y[x],x])^2+a x^2+b*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\text {RootSum}\left [\text {$\#$1}^4-\text {$\#$1}^3 b+2 \text {$\#$1}^2 a+\text {$\#$1} a b+a^2\&,\frac {2 \text {$\#$1}^3 \log \left (\text {$\#$1} x-\sqrt {-a x^2-b y(x)}+\sqrt {-b y(x)}\right )-2 \text {$\#$1}^3 \log (x)-\text {$\#$1}^2 b \log \left (\text {$\#$1} x-\sqrt {-a x^2-b y(x)}+\sqrt {-b y(x)}\right )+\text {$\#$1}^2 b \log (x)+2 \text {$\#$1} a \log \left (\text {$\#$1} x-\sqrt {-a x^2-b y(x)}+\sqrt {-b y(x)}\right )+a b \log \left (\text {$\#$1} x-\sqrt {-a x^2-b y(x)}+\sqrt {-b y(x)}\right )-2 \text {$\#$1} a \log (x)-a b \log (x)}{4 \text {$\#$1}^3-3 \text {$\#$1}^2 b+4 \text {$\#$1} a+a b}\&\right ]-\log \left (\sqrt {-b y(x)} \sqrt {-a x^2-b y(x)}+b y(x)\right )+\frac {1}{2} \log (y(x))+2 \log (x)=c_1,y(x)\right ]\\ \text {Solve}\left [\text {RootSum}\left [\text {$\#$1}^4+\text {$\#$1}^3 b+2 \text {$\#$1}^2 a-\text {$\#$1} a b+a^2\&,\frac {-2 \text {$\#$1}^3 \log \left (\text {$\#$1} x-\sqrt {-a x^2-b y(x)}+\sqrt {-b y(x)}\right )+2 \text {$\#$1}^3 \log (x)-\text {$\#$1}^2 b \log \left (\text {$\#$1} x-\sqrt {-a x^2-b y(x)}+\sqrt {-b y(x)}\right )+\text {$\#$1}^2 b \log (x)-2 \text {$\#$1} a \log \left (\text {$\#$1} x-\sqrt {-a x^2-b y(x)}+\sqrt {-b y(x)}\right )+a b \log \left (\text {$\#$1} x-\sqrt {-a x^2-b y(x)}+\sqrt {-b y(x)}\right )+2 \text {$\#$1} a \log (x)-a b \log (x)}{-4 \text {$\#$1}^3-3 \text {$\#$1}^2 b-4 \text {$\#$1} a+a b}\&\right ]-\log \left (\sqrt {-b y(x)} \sqrt {-a x^2-b y(x)}+b y(x)\right )+\frac {1}{2} \log (y(x))+2 \log (x)=c_1,y(x)\right ] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*x**2 + b*y(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-a*x**2 - b*y(x)) + Derivative(y(x), x) cannot be solved b