23.2.334 problem 376

Internal problem ID [5689]
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 : 376
Date solved : Tuesday, September 30, 2025 at 01:57:53 PM
CAS classification : [_quadrature]

\begin{align*} \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }&=x \end{align*}

✓ Maple. Time used: 0.061 (sec). Leaf size: 113

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

\begin{align*} y &= \frac {a \,x^{2}+x \sqrt {a^{2}+x^{2}-1}+\left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )+2 c_1 \right ) \left (a +1\right ) \left (a -1\right )}{2 a^{2}-2} \\ y &= \frac {a \,x^{2}-x \sqrt {a^{2}+x^{2}-1}-\left (a +1\right ) \left (a -1\right ) \left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )-2 c_1 \right )}{2 a^{2}-2} \\ \end{align*}

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 109

ode=Sqrt[1+(D[y[x],x])^2]+ a*D[y[x],x]==x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (\frac {x \left (a x-\sqrt {a^2+x^2-1}\right )}{a^2-1}-\text {arctanh}\left (\frac {x}{\sqrt {a^2+x^2-1}}\right )\right )+c_1\\ y(x)&\to \frac {1}{2} \left (\text {arctanh}\left (\frac {x}{\sqrt {a^2+x^2-1}}\right )+\frac {x \left (\sqrt {a^2+x^2-1}+a x\right )}{a^2-1}\right )+c_1 \end{align*}

✓ Sympy. Time used: 1.299 (sec). Leaf size: 134

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*Derivative(y(x), x) - x + sqrt(Derivative(y(x), x)**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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