82.46.1 problem Ex. 1

Internal problem ID [18901]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 101
Problem number : Ex. 1
Date solved : Monday, March 31, 2025 at 06:21:23 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 71

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

\begin{align*} y &= \frac {c_1 \ln \left (x +\sqrt {x^{2}+c_1}\right )+x \sqrt {x^{2}+c_1}+2 c_2 a}{2 a} \\ y &= \frac {-c_1 \ln \left (x +\sqrt {x^{2}+c_1}\right )-x \sqrt {x^{2}+c_1}+2 c_2 a}{2 a} \\ \end{align*}

✓ Mathematica. Time used: 28.58 (sec). Leaf size: 106

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

\begin{align*} y(x)\to -\frac {x \sqrt {x^2+2 a^2 c_1}}{2 a}-a c_1 \log \left (x+\sqrt {x^2+2 a^2 c_1}\right )+c_2 \\ y(x)\to \frac {x \sqrt {x^2+2 a^2 c_1}}{2 a}+a c_1 \log \left (x+\sqrt {x^2+2 a^2 c_1}\right )+c_2 \\ \end{align*}

✓ Sympy. Time used: 3.668 (sec). Leaf size: 82

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

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