83.19.3 problem 3

Internal problem ID [19231]
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 : Tuesday, January 28, 2025 at 01:14:47 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.199 (sec). Leaf size: 66

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

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

✓ Solution by Mathematica

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

DSolve[y[x]==x+a*ArcTan[D[y[x],x]],y[x],x,IncludeSingularSolutions -> True]
 

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