Internal problem ID [4104]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 30
Problem number: 867.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {x {y^{\prime }}^{2}-2 y y^{\prime }=-x a} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 33
dsolve(x*diff(y(x),x)^2-2*y(x)*diff(y(x),x)+a*x = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \sqrt {a}\, x y \left (x \right ) = -\sqrt {a}\, x y \left (x \right ) = \frac {\left (\frac {x^{2}}{c_{1}^{2}}+a \right ) c_{1}}{2} \end{align*}
✓ Solution by Mathematica
Time used: 16.916 (sec). Leaf size: 400
DSolve[x (y'[x])^2-2 y[x] y'[x]+a x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {a} x \tan (c_1-i \log (x))}{\sqrt {\sec ^2(c_1-i \log (x))}} y(x)\to \frac {\sqrt {a} x \tan (c_1-i \log (x))}{\sqrt {\sec ^2(c_1-i \log (x))}} y(x)\to -\frac {\sqrt {a} x \tan (i \log (x)+c_1)}{\sqrt {\sec ^2(i \log (x)+c_1)}} y(x)\to \frac {\sqrt {a} x \tan (i \log (x)+c_1)}{\sqrt {\sec ^2(i \log (x)+c_1)}} y(x)\to -\sqrt {a} x y(x)\to \sqrt {a} x y(x)\to \frac {i \sqrt {a} e^{2 i \text {Interval}[\{0,\pi \}]} \left (e^{2 i \text {Interval}[\{0,\pi \}]} \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}-x^4 \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}\right )}{2 x} y(x)\to \frac {i \sqrt {a} e^{2 i \text {Interval}[\{0,\pi \}]} \left (x^4 \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}-e^{2 i \text {Interval}[\{0,\pi \}]} \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}\right )}{2 x} \end{align*}