Internal problem ID [3673]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 32
Problem number: 947.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class C], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y \left (y^{\prime }\right )^{2}-\left (-2 b x +a \right ) y^{\prime }-b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.656 (sec). Leaf size: 203
dsolve(y(x)*diff(y(x),x)^2-(-2*b*x+a)*diff(y(x),x)-b*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \sqrt {\frac {c_{1} b +\sqrt {4 b^{3} x^{2} c_{1}-4 a \,b^{2} x c_{1}+a^{2} b c_{1}}}{b}} \\ y \relax (x ) = -\sqrt {\frac {c_{1} b +\sqrt {4 b^{3} x^{2} c_{1}-4 a \,b^{2} x c_{1}+a^{2} b c_{1}}}{b}} \\ y \relax (x ) = \sqrt {-\frac {-c_{1} b +\sqrt {4 b^{3} x^{2} c_{1}-4 a \,b^{2} x c_{1}+a^{2} b c_{1}}}{b}} \\ y \relax (x ) = -\sqrt {-\frac {-c_{1} b +\sqrt {4 b^{3} x^{2} c_{1}-4 a \,b^{2} x c_{1}+a^{2} b c_{1}}}{b}} \\ y \relax (x ) = \sqrt {-b \,x^{2}+a x +c_{1}} \\ y \relax (x ) = -\sqrt {-b \,x^{2}+a x +c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.21 (sec). Leaf size: 409
DSolve[y[x] (y'[x])^2-(a-2 b x)y'[x]-b y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {\left (b-\sqrt {b^2}\right ) \log (y(x))}{b}-\frac {-b \log \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}-a-2 \sqrt {b^2} x\right )+\sqrt {b^2} \log \left (b \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}-a-2 \sqrt {b^2} x\right )\right )-\left (\sqrt {b^2}+b\right ) \log \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}+a-2 \sqrt {b^2} x\right )}{2 \sqrt {b^2}}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {-b \log \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}-a-2 \sqrt {b^2} x\right )+\sqrt {b^2} \log \left (b \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}-a-2 \sqrt {b^2} x\right )\right )-\left (\sqrt {b^2}+b\right ) \log \left (\sqrt {a^2-4 a b x+4 b \left (b x^2+y(x)^2\right )}+a-2 \sqrt {b^2} x\right )}{2 \sqrt {b^2}}+\frac {\left (\sqrt {b^2}+b\right ) \log (y(x))}{b}=c_1,y(x)\right ] \\ y(x)\to -\frac {i (2 b x-a)}{2 \sqrt {b}} \\ y(x)\to \frac {i (2 b x-a)}{2 \sqrt {b}} \\ \end{align*}