29.9.1 problem 241

Internal problem ID [4841]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 9
Problem number : 241
Date solved : Monday, January 27, 2025 at 09:42:18 AM
CAS classification : [_separable]

\begin{align*} 2 x y^{\prime }+4 y+a +\sqrt {a^{2}-4 b -4 c y}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 34

dsolve(2*x*diff(y(x),x)+4*y(x)+a+sqrt(a^2-4*b-4*c*y(x)) = 0,y(x), singsol=all)
 

\[ \ln \left (x \right )+2 \left (\int _{}^{y \left (x \right )}\frac {1}{4 \textit {\_a} +a +\sqrt {-4 \textit {\_a} c +a^{2}-4 b}}d \textit {\_a} \right )+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.883 (sec). Leaf size: 176

DSolve[2 x D[y[x],x]+4 y[x]+a +Sqrt[a^2-4 b- 4 c y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{4} \left (\log \left (-c \left (\sqrt {a^2-4 (\text {$\#$1} c+b)}+4 \text {$\#$1}+a\right )\right )-\frac {2 c \text {arctanh}\left (\frac {2 \sqrt {a^2-4 (\text {$\#$1} c+b)}-c}{\sqrt {4 a^2+4 a c-16 b+c^2}}\right )}{\sqrt {4 a^2+4 a c-16 b+c^2}}\right )\&\right ]\left [-\frac {\log (x)}{2}+c_1\right ] \\ y(x)\to \frac {1}{8} \left (-\sqrt {(2 a+c)^2-16 b}-2 a-c\right ) \\ y(x)\to \frac {1}{8} \left (\sqrt {(2 a+c)^2-16 b}-2 a-c\right ) \\ \end{align*}