83.19.3 problem 3

Internal problem ID [19160]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (B) at page 55
Problem number : 3
Date solved : Monday, March 31, 2025 at 06:50:24 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y&=x +a \arctan \left (y^{\prime }\right ) \end{align*}

✓ Maple. Time used: 0.103 (sec). Leaf size: 66

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

\begin{align*} y &= x +\frac {a \pi }{4} \\ -\frac {a \ln \left (\tan \left (\frac {y-x}{a}\right )-1\right )}{2}+\frac {a \ln \left (\sec \left (\frac {y-x}{a}\right )^{2}\right )}{4}+\frac {a \arctan \left (\tan \left (\frac {y-x}{a}\right )\right )}{2}-c_1 +x &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.305 (sec). Leaf size: 89

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

\[ \text {Solve}\left [\left (\frac {1}{4}-\frac {i}{4}\right ) a \log \left (-\tan \left (\frac {x}{a}-\frac {y(x)}{a}\right )+i\right )+\left (\frac {1}{4}+\frac {i}{4}\right ) a \log \left (\tan \left (\frac {x}{a}-\frac {y(x)}{a}\right )+i\right )-\frac {1}{2} a \log \left (\tan \left (\frac {x}{a}-\frac {y(x)}{a}\right )+1\right )+y(x)=c_1,y(x)\right ] \]

✓ Sympy. Time used: 14.985 (sec). Leaf size: 42

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

\[ C_{1} - \frac {a \log {\left (\tan {\left (\frac {x - y{\left (x \right )}}{a} \right )} + 1 \right )}}{2} + \frac {a \log {\left (\tan ^{2}{\left (\frac {x - y{\left (x \right )}}{a} \right )} + 1 \right )}}{4} + \frac {x}{2} + \frac {y{\left (x \right )}}{2} = 0 \]