29.14.11 problem 392

Internal problem ID [4990]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 392
Date solved : Sunday, March 30, 2025 at 04:28:40 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 36

ode:=diff(y(x),x)*(a^2+x^2)^(1/2)+x+y(x) = (a^2+x^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {a^{2} \ln \left (x +\sqrt {a^{2}+x^{2}}\right )+c_1}{x +\sqrt {a^{2}+x^{2}}} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 42

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

\[ y(x)\to \frac {a^2 \log \left (\sqrt {a^2+x^2}+x\right )+c_1}{\sqrt {a^2+x^2}+x} \]

✓ Sympy. Time used: 5.452 (sec). Leaf size: 63

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

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