29.28.1 problem 798

Internal problem ID [5382]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 28
Problem number : 798
Date solved : Sunday, March 30, 2025 at 08:04:03 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 53

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

\begin{align*} y &= \frac {x^{2} \sqrt {a^{2}+4 c b}}{4}-\frac {x^{2} a}{4}+c_1 \\ y &= -\frac {x^{2} \sqrt {a^{2}+4 c b}}{4}-\frac {x^{2} a}{4}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 59

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

\begin{align*} y(x)\to \frac {1}{4} x^2 \left (\sqrt {a^2+4 b c}-a\right )+c_1 \\ y(x)\to -\frac {1}{4} x^2 \left (\sqrt {a^2+4 b c}+a\right )+c_1 \\ \end{align*}

✓ Sympy. Time used: 0.289 (sec). Leaf size: 54

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

\[ \left [ y{\left (x \right )} = C_{1} - \frac {a x^{2}}{4} + \frac {x^{2} \sqrt {a^{2} + 4 b c}}{4}, \ y{\left (x \right )} = C_{1} - \frac {a x^{2}}{4} - \frac {x^{2} \sqrt {a^{2} + 4 b c}}{4}\right ] \]