31.6.6 problem 6

Internal problem ID [5755]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 6
Date solved : Sunday, March 30, 2025 at 10:08:12 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 80

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

\begin{align*} y &= \frac {\left (a^{2}+4 b x \right )^{{3}/{2}}+12 c_1 \,b^{2}-6 a x b}{12 b^{2}} \\ y &= \frac {12 c_1 \,b^{2}-a^{2} \sqrt {a^{2}+4 b x}-6 a x b -4 b \sqrt {a^{2}+4 b x}\, x}{12 b^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 74

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

\begin{align*} y(x)\to \frac {\left (a^2+4 b x\right )^{3/2}-6 a b x+12 b^2 c_1}{12 b^2} \\ y(x)\to -\frac {\frac {\left (a^2+4 b x\right )^{3/2}}{6 b}+a x}{2 b}+c_1 \\ \end{align*}

✓ Sympy. Time used: 0.452 (sec). Leaf size: 112

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

\[ \left [ y{\left (x \right )} = C_{1} - \frac {\begin {cases} \frac {a^{2} \sqrt {a^{2} + 4 b x}}{6 b} + a x + \frac {2 x \sqrt {a^{2} + 4 b x}}{3} & \text {for}\: b \neq 0 \\x \left (a + \sqrt {a^{2}}\right ) & \text {otherwise} \end {cases}}{2 b}, \ y{\left (x \right )} = C_{1} - \frac {\begin {cases} - \frac {a^{2} \sqrt {a^{2} + 4 b x}}{6 b} + a x - \frac {2 x \sqrt {a^{2} + 4 b x}}{3} & \text {for}\: b \neq 0 \\x \left (a - \sqrt {a^{2}}\right ) & \text {otherwise} \end {cases}}{2 b}\right ] \]