Internal problem ID [8648]
Internal file name [OUTPUT/7581_Sunday_June_05_2022_11_07_59_PM_92628403/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 312.
ODE order: 1.
ODE degree: 1.
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.
\[ \boxed {\left (\frac {y^{2}}{b}+\frac {x^{2}}{a}\right ) \left (y y^{\prime }+x \right )+\frac {\left (a -b \right ) \left (y y^{\prime }-x \right )}{a +b}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{3} y^{\prime } a^{2}+y^{3} y^{\prime } a b +y y^{\prime } a b \,x^{2}+y y^{\prime } b^{2} x^{2}+y^{2} a^{2} x +y^{2} a b x +y y^{\prime } a^{2} b -y y^{\prime } a \,b^{2}+a b \,x^{3}+b^{2} x^{3}-a^{2} b x +a \,b^{2} x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{2} a^{2} x -y^{2} a b x -a b \,x^{3}-b^{2} x^{3}+a^{2} b x -a \,b^{2} x}{y^{3} a^{2}+y^{3} a b +y a b \,x^{2}+y b^{2} x^{2}+y a^{2} b -y a \,b^{2}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 2`[0, (-a*y^2-b*x^2+a*b)/y/(a^2*y^2+a*b*x^2+a*b*y^2+b^2*x^2+a^2*b-a*b^2)]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 244
dsolve((y(x)^2/b+x^2/a)*(y(x)*diff(y(x),x)+x)+(a-b)/(a+b)*(y(x)*diff(y(x),x)-x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\sqrt {a \left ({\mathrm e}^{\frac {-2 \operatorname {LambertW}\left (\frac {\left (a +b \right ) {\mathrm e}^{\frac {\left (-x^{2}-b \right ) a^{2}+\left (-b^{2}-2 c_{1} \right ) a +b^{2} x^{2}}{2 a^{2} b}}}{2 a^{2} b}\right ) a^{2} b +\left (-x^{2}-b \right ) a^{2}+\left (-b^{2}-2 c_{1} \right ) a +b^{2} x^{2}}{2 a^{2} b}}+b \left (-x^{2}+a \right )\right )}}{a} \\ y \left (x \right ) &= -\frac {\sqrt {a \left ({\mathrm e}^{\frac {-2 \operatorname {LambertW}\left (\frac {\left (a +b \right ) {\mathrm e}^{\frac {\left (-x^{2}-b \right ) a^{2}+\left (-b^{2}-2 c_{1} \right ) a +b^{2} x^{2}}{2 a^{2} b}}}{2 a^{2} b}\right ) a^{2} b +\left (-x^{2}-b \right ) a^{2}+\left (-b^{2}-2 c_{1} \right ) a +b^{2} x^{2}}{2 a^{2} b}}+b \left (-x^{2}+a \right )\right )}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.984 (sec). Leaf size: 178
DSolve[((a - b)*(-x + y[x]*y'[x]))/(a + b) + (x^2/a + y[x]^2/b)*(x + y[x]*y'[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {b} \sqrt {(a+b) \left (a-x^2\right )+2 a^2 W\left (\frac {c_1 (a+b) \exp \left (-\frac {(a+b) \left (a \left (b+x^2\right )-b x^2\right )}{2 a^2 b}\right )}{2 a^3 b^2}\right )}}{\sqrt {a} \sqrt {a+b}} \\ y(x)\to \frac {\sqrt {b} \sqrt {(a+b) \left (a-x^2\right )+2 a^2 W\left (\frac {c_1 (a+b) \exp \left (-\frac {(a+b) \left (a \left (b+x^2\right )-b x^2\right )}{2 a^2 b}\right )}{2 a^3 b^2}\right )}}{\sqrt {a} \sqrt {a+b}} \\ \end{align*}