1.486 problem 489

Internal problem ID [9468]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 489
Date solved : Thursday, October 17, 2024 at 02:41:47 PM
CAS classification : [_rational]

Solve

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

Solving for the derivative gives these ODE’s to solve

\begin{align*} \tag{1} y^{\prime }&=\frac {-x +\sqrt {x^{2}-a y^{2}-b x -c}}{y} \\ \tag{2} y^{\prime }&=\frac {-x -\sqrt {x^{2}-a y^{2}-b x -c}}{y} \\ \end{align*}

Now each of the above is solved separately.

Solving Eq. (1)

Solving Eq. (2)

1.486.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right )^{2} \left (\frac {d}{d x}y \left (x \right )\right )^{2}+2 x y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+a y \left (x \right )^{2}+b x +c =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d x}y \left (x \right )=\frac {-x -\sqrt {x^{2}-a y \left (x \right )^{2}-b x -c}}{y \left (x \right )}, \frac {d}{d x}y \left (x \right )=\frac {-x +\sqrt {x^{2}-a y \left (x \right )^{2}-b x -c}}{y \left (x \right )}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=\frac {-x -\sqrt {x^{2}-a y \left (x \right )^{2}-b x -c}}{y \left (x \right )} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=\frac {-x +\sqrt {x^{2}-a y \left (x \right )^{2}-b x -c}}{y \left (x \right )} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]

1.486.2 Maple trace

`Methods for first order ODEs: 
-> Solving 1st order ODE of high degree, 1st attempt 
trying 1st order WeierstrassP solution for high degree ODE 
trying 1st order WeierstrassPPrime solution for high degree ODE 
trying 1st order JacobiSN solution for high degree ODE 
trying 1st order ODE linearizable_by_differentiation 
trying differential order: 1; missing variables 
trying simple symmetries for implicit equations 
Successful isolation of dy/dx: 2 solutions were found. Trying to solve each resulting ODE. 
   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying homogeneous types: 
   trying exact 
   Looking for potential symmetries 
   trying an equivalence to an Abel ODE 
   trying 1st order ODE linearizable_by_differentiation 
-> Solving 1st order ODE of high degree, Lie methods, 1st trial 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 4`[(2*a*x-b+2*x)/b, 1/2*(4*a^2*y^2+2*a*b*x+4*a*y^2+4*a*c+b^2+4*c)/a/b/y]
 

1.486.3 Maple dsolve solution

Solving time : 1.053 (sec)
Leaf size : 363

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

\begin{align*} y &= -\frac {2 \sqrt {a \left (a \left (a x -\frac {1}{2} b +x \right )^{2} \left (a +1\right )^{2} \operatorname {RootOf}\left (-2 b \ln \left (2 a x -b +2 x \right )-b \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a} \,a^{2}+\sqrt {-{\mathrm e}^{\frac {4 a}{b}} {\mathrm e}^{\frac {4}{b}} \left (4 \textit {\_a} \,a^{3}+8 \textit {\_a} \,a^{2}+4 \textit {\_a} a -1\right )}\, {\mathrm e}^{-\frac {2 \left (a +1\right )}{b}}+8 \textit {\_a} a +4 \textit {\_a} +1}{\textit {\_a} \left (4 \textit {\_a} \,a^{2}+8 \textit {\_a} a +4 \textit {\_a} +a +2\right )}d \textit {\_a} \right )+4 c_{1} a +4 c_{1} \right )+\frac {\left (-b x -c \right ) a^{2}}{4}+\frac {\left (-\frac {b x}{2}-c \right ) a}{2}-\frac {b^{2}}{16}-\frac {c}{4}\right )}}{a \left (a +1\right )} \\ y &= \frac {2 \sqrt {a \left (a \left (a x -\frac {1}{2} b +x \right )^{2} \left (a +1\right )^{2} \operatorname {RootOf}\left (-2 b \ln \left (2 a x -b +2 x \right )-b \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a} \,a^{2}+\sqrt {-{\mathrm e}^{\frac {4 a}{b}} {\mathrm e}^{\frac {4}{b}} \left (4 \textit {\_a} \,a^{3}+8 \textit {\_a} \,a^{2}+4 \textit {\_a} a -1\right )}\, {\mathrm e}^{-\frac {2 \left (a +1\right )}{b}}+8 \textit {\_a} a +4 \textit {\_a} +1}{\textit {\_a} \left (4 \textit {\_a} \,a^{2}+8 \textit {\_a} a +4 \textit {\_a} +a +2\right )}d \textit {\_a} \right )+4 c_{1} a +4 c_{1} \right )+\frac {\left (-b x -c \right ) a^{2}}{4}+\frac {\left (-\frac {b x}{2}-c \right ) a}{2}-\frac {b^{2}}{16}-\frac {c}{4}\right )}}{a \left (a +1\right )} \\ \end{align*}

1.486.4 Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

DSolve[{c + b*x + a*y[x]^2 + 2*x*y[x]*D[y[x],x] + y[x]^2*D[y[x],x]^2==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

Timed out