Internal problem ID [4146]
Internal file name [OUTPUT/3639_Sunday_June_05_2022_09_53_13_AM_94420098/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 910.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational]
Unable to solve or complete the solution.
\[ \boxed {x^{2} {y^{\prime }}^{2}+x \left (x^{2}+x y-2 y\right ) y^{\prime }+\left (1-x \right ) \left (x^{2}-y\right ) y=0} \] Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {-\frac {x^{2}}{2}-\frac {x y}{2}+y+\frac {\sqrt {y^{2} x^{2}+6 y x^{3}+x^{4}-8 y^{2} x -8 y x^{2}+8 y^{2}}}{2}}{x} \tag {1} \\ y^{\prime }&=\frac {-\frac {x^{2}}{2}-\frac {x y}{2}+y-\frac {\sqrt {y^{2} x^{2}+6 y x^{3}+x^{4}-8 y^{2} x -8 y x^{2}+8 y^{2}}}{2}}{x} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Unable to determine ODE type.
Unable to determine ODE type.
Solving equation (2)
Unable to determine ODE type.
Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} {y^{\prime }}^{2}+x \left (x^{2}+x y-2 y\right ) y^{\prime }+\left (1-x \right ) \left (x^{2}-y\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {-\frac {x^{2}}{2}-\frac {x y}{2}+y-\frac {\sqrt {y^{2} x^{2}+6 y x^{3}+x^{4}-8 y^{2} x -8 y x^{2}+8 y^{2}}}{2}}{x}, y^{\prime }=\frac {-\frac {x^{2}}{2}-\frac {x y}{2}+y+\frac {\sqrt {y^{2} x^{2}+6 y x^{3}+x^{4}-8 y^{2} x -8 y x^{2}+8 y^{2}}}{2}}{x}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {-\frac {x^{2}}{2}-\frac {x y}{2}+y-\frac {\sqrt {y^{2} x^{2}+6 y x^{3}+x^{4}-8 y^{2} x -8 y x^{2}+8 y^{2}}}{2}}{x} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {-\frac {x^{2}}{2}-\frac {x y}{2}+y+\frac {\sqrt {y^{2} x^{2}+6 y x^{3}+x^{4}-8 y^{2} x -8 y x^{2}+8 y^{2}}}{2}}{x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
✗ Solution by Maple
dsolve(x^2*diff(y(x),x)^2+x*(x^2+x*y(x)-2*y(x))*diff(y(x),x)+(1-x)*(x^2-y(x))*y(x) = 0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x^2 (y'[x])^2+x(x^2+x y[x]-2 y[x])y'[x]+(1-x)(x^2-y[x])y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved