24.1.23 problem 5(d)

Internal problem ID [4212]
Book : Elementary Differential equations, Chaundy, 1969
Section : Exercises 3, page 60
Problem number : 5(d)
Date solved : Monday, January 27, 2025 at 08:42:27 AM
CAS classification : [_linear]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 114

dsolve(sqrt((x+a)*(x+b))*diff(y(x),x)+y(x)=sqrt(x+a)-sqrt(x+b),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {2 \left (\left (2 a +2 x \right ) \sqrt {x +a}+\left (-2 b -2 x \right ) \sqrt {x +b}+3 c_{1} \right ) \sqrt {\left (x +a \right ) \left (x +b \right )}+6 \left (x +b \right ) \left (-\frac {b}{3}+a +\frac {2 x}{3}\right ) \sqrt {x +a}+2 \sqrt {x +b}\, \left (x +a \right ) \left (-2 x +a -3 b \right )}{\sqrt {\left (x +a \right ) \left (x +b \right )}\, \left (3 a +3 b +6 x +6 \sqrt {\left (x +a \right ) \left (x +b \right )}\right )} \]

✓ Solution by Mathematica

Time used: 1.687 (sec). Leaf size: 145

DSolve[Sqrt[(x+a)*(x+b)]*D[y[x],x]+y[x]==Sqrt[x+a]-Sqrt[x+b],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (-\frac {2 \sqrt {a+x} \sqrt {b+x} \text {arctanh}\left (\frac {\sqrt {b+x}}{\sqrt {a+x}}\right )}{\sqrt {(a+x) (b+x)}}\right ) \left (\int _1^x\frac {\exp \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {b+K[1]}}{\sqrt {a+K[1]}}\right ) \sqrt {a+K[1]} \sqrt {b+K[1]}}{\sqrt {(a+K[1]) (b+K[1])}}\right ) \left (\sqrt {a+K[1]}-\sqrt {b+K[1]}\right )}{\sqrt {(a+K[1]) (b+K[1])}}dK[1]+c_1\right ) \]