ODE
\[ a y'(x)+b y(x)+y'(x)^2=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y\)
Mathematica ✓
cpu = 0.313858 (sec), leaf count = 110
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {$\#$1} b}+a \log \left (\sqrt {a^2-4 \text {$\#$1} b}-a\right )}{2 b}\& \right ]\left [c_1+\frac {x}{2}\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {$\#$1} b}-a \log \left (\sqrt {a^2-4 \text {$\#$1} b}+a\right )}{2 b}\& \right ]\left [c_1-\frac {x}{2}\right ]\right \}\right \}\]
Maple ✓
cpu = 0.043 (sec), leaf count = 141
\[ \left \{ {\frac {1}{2\,b} \left ( a\ln \left ( y \left ( x \right ) \right ) + \left ( -\ln \left ( a+\sqrt {{a}^{2}-4\,by \left ( x \right ) } \right ) +\ln \left ( -a+\sqrt {{a}^{2}-4\,by \left ( x \right ) } \right ) \right ) a+ \left ( 2\,x-2\,{\it \_C1} \right ) b+2\,\sqrt {{a}^{2}-4\,by \left ( x \right ) } \right ) }=0,{\frac {1}{2\,b} \left ( a\ln \left ( y \left ( x \right ) \right ) + \left ( \ln \left ( a+\sqrt {{a}^{2}-4\,by \left ( x \right ) } \right ) -\ln \left ( -a+\sqrt {{a}^{2}-4\,by \left ( x \right ) } \right ) \right ) a+ \left ( 2\,x-2\,{\it \_C1} \right ) b-2\,\sqrt {{a}^{2}-4\,by \left ( x \right ) } \right ) }=0 \right \} \] Mathematica raw input
DSolve[b*y[x] + a*y'[x] + y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-(a*Log[-a + Sqrt[a^2 - 4*b*#1]] + Sqrt[a^2 - 4*b*#1])/(2*b) & ][x/2 + C[1]]}, {y[x] -> InverseFunction[-(-(a*Log[a + Sqrt[a^2 - 4*b*#1]]) + Sqrt[a^2 - 4*b*#1])/(2*b) & ][-x/2 + C[1]]}}
Maple raw input
dsolve(diff(y(x),x)^2+a*diff(y(x),x)+b*y(x) = 0, y(x),'implicit')
Maple raw output
1/2*(a*ln(y(x))+(ln(a+(a^2-4*b*y(x))^(1/2))-ln(-a+(a^2-4*b*y(x))^(1/2)))*a+(2*x-2*_C1)*b-2*(a^2-4*b*y(x))^(1/2))/b = 0, 1/2*(a*ln(y(x))+(-ln(a+(a^2-4*b*y(x))^(1/2))+ln(-a+(a^2-4*b*y(x))^(1/2)))*a+(2*x-2*_C1)*b+2*(a^2-4*b*y(x))^(1/2))/b = 0