29.12.7 problem 326

Internal problem ID [4926]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 12
Problem number : 326
Date solved : Monday, January 27, 2025 at 09:51:30 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} \left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (x +y-a \right ) \left (x +y-b \right )+y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 54

dsolve((x-a)*(x-b)*diff(y(x),x)+k*(x+y(x)-a)*(x+y(x)-b)+y(x)^2 = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {k \left (\left (-x +a \right )^{k +1}+c_{1} \left (-x +b \right )^{k} \left (-x +b \right )\right )}{\left (k +1\right ) \left (c_{1} \left (-x +b \right )^{k}+\left (-x +a \right )^{k}\right )} \]

✓ Solution by Mathematica

Time used: 60.329 (sec). Leaf size: 99

DSolve[(x-a)(x-b)D[y[x],x]+k(x+y[x]-a)(x+y[x]-b)+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} \left (\frac {k (a+b-2 x)}{k+1}+\sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} \tan \left (\frac {(k+1) \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} (\log (x-b)-\log (x-a))}{2 (a-b)}+c_1\right )\right ) \]