4.5.44 \(\sqrt {a^2-4 b-4 c y(x)}+a+2 x y'(x)+4 y(x)=0\)

ODE
\[ \sqrt {a^2-4 b-4 c y(x)}+a+2 x y'(x)+4 y(x)=0 \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 2.02221 (sec), leaf count = 357

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

Maple
cpu = 0.032 (sec), leaf count = 34

\[ \left \{ \ln \relax (x ) +\int ^{y \relax (x ) }\!2\, \left (4\,{\it \_a}+a+\sqrt {-4\,{\it \_a}\,c+{a}^{2}-4\,b} \right ) ^{-1}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[a + 4*y[x] + Sqrt[a^2 - 4*b - 4*c*y[x]] + 2*x*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[((-2*c*ArcTan[(2*a + c + 8*#1)/Sqrt[-4*a^2 + 16*b - 4*
a*c - c^2]])/Sqrt[-4*a^2 + 16*b - 4*a*c - c^2] - (2*Sqrt[2*a^2 - 8*b + 2*a*c + c
*(c + Sqrt[4*a^2 - 16*b + 4*a*c + c^2])]*ArcTanh[(Sqrt[2]*Sqrt[a^2 - 4*(b + c*#1
)])/Sqrt[2*a^2 - 8*b + 2*a*c + c*(c + Sqrt[4*a^2 - 16*b + 4*a*c + c^2])]])/Sqrt[
2*a^2 - 8*b + 2*a*c + c^2/2] + (2*Sqrt[2*a^2 - 8*b + 2*a*c + c^2 - c*Sqrt[4*a^2 
- 16*b + 4*a*c + c^2]]*ArcTanh[Sqrt[a^2 - 4*(b + c*#1)]/Sqrt[a^2 + a*c + (-8*b +
 c*(c - Sqrt[4*a^2 - 16*b + 4*a*c + c^2]))/2]])/Sqrt[2*a^2 - 8*b + 2*a*c + c^2/2
] + Log[b + #1*(2*a + c + 4*#1)])/8 & ][C[1] - Log[x]/2]}}

Maple raw input

dsolve(2*x*diff(y(x),x)+4*y(x)+a+(a^2-4*b-4*c*y(x))^(1/2) = 0, y(x),'implicit')

Maple raw output

ln(x)+Intat(2/(4*_a+a+(-4*_a*c+a^2-4*b)^(1/2)),_a = y(x))+_C1 = 0