Internal problem ID [9175]
Internal file name [OUTPUT/8110_Monday_June_06_2022_01_47_26_AM_51604843/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 841.
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.
\[ \boxed {y^{\prime }-\frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{\frac {3}{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{\frac {3}{2}} b \,x^{2}+a^{\frac {5}{2}} y^{4}}{a \,x^{2} y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -a^{\frac {5}{2}} y^{4}+2 y^{2} a^{\frac {3}{2}} b \,x^{2}-b^{2} x^{4} \sqrt {a}-2 c y^{2} a^{\frac {3}{2}}+y^{\prime } y a \,x^{2}+2 c b \,x^{2} \sqrt {a}-b \,x^{3}-c^{2} \sqrt {a}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{\frac {3}{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{\frac {3}{2}} b \,x^{2}+a^{\frac {5}{2}} y^{4}}{a \,x^{2} y} \end {array} \]
Maple trace Kovacic algorithm successful
`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 -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -4*(a*b^2*x^4+b*x^3*a^(1/2)-2*b*c*x^2*a+a*c^2)*y(x)/x^4-2*(2*a*x^2*b-2*a* Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) <- Kovacics algorithm successful <- differential order: 1; linearization to 2nd order successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 97
dsolve(diff(y(x),x) = (b*x^3+c^2*a^(1/2)-2*c*b*x^2*a^(1/2)+2*c*y(x)^2*a^(3/2)+b^2*x^4*a^(1/2)-2*y(x)^2*a^(3/2)*b*x^2+a^(5/2)*y(x)^4)/a/x^2/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {2 \sqrt {\left (c_{1} x +1\right ) \left (\left (c_{1} x +1\right ) \left (b \,x^{2}-c \right ) \sqrt {a}+\frac {x}{2}\right ) a^{\frac {3}{2}}}}{a^{\frac {3}{2}} \left (2 c_{1} x +2\right )} \\ y \left (x \right ) &= \frac {\sqrt {\left (c_{1} x +1\right ) \left (\left (c_{1} x +1\right ) \left (b \,x^{2}-c \right ) \sqrt {a}+\frac {x}{2}\right ) a^{\frac {3}{2}}}}{a^{\frac {3}{2}} \left (c_{1} x +1\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 9.413 (sec). Leaf size: 390
DSolve[y'[x] == (Sqrt[a]*c^2 - 2*Sqrt[a]*b*c*x^2 + b*x^3 + Sqrt[a]*b^2*x^4 + 2*a^(3/2)*c*y[x]^2 - 2*a^(3/2)*b*x^2*y[x]^2 + a^(5/2)*y[x]^4)/(a*x^2*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-2 a^{5/2} \left (c-b x^2\right )+4 a^3 b x \left (b x^2-c\right )+a^2 x+4 \sqrt {a} b c_1 \left (b x^2-c\right )+2 b c_1 x}}{\sqrt {2} \sqrt {2 a^{3/2} b c_1+a^{7/2}+2 a^4 b x}} \\ y(x)\to \frac {\sqrt {-2 a^{5/2} \left (c-b x^2\right )+4 a^3 b x \left (b x^2-c\right )+a^2 x+4 \sqrt {a} b c_1 \left (b x^2-c\right )+2 b c_1 x}}{\sqrt {2} \sqrt {2 a^{3/2} b c_1+a^{7/2}+2 a^4 b x}} \\ y(x)\to -\frac {\sqrt {-b \left (x-2 \sqrt {a} \left (c-b x^2\right )\right )}}{\sqrt {2} \sqrt {-a^{3/2} b}} \\ y(x)\to \frac {\sqrt {-b \left (x-2 \sqrt {a} \left (c-b x^2\right )\right )}}{\sqrt {2} \sqrt {-a^{3/2} b}} \\ y(x)\to -\frac {\sqrt {b \left (x-2 \sqrt {a} \left (c-b x^2\right )\right )}}{\sqrt {2} \sqrt {a^{3/2} b}} \\ y(x)\to \frac {\sqrt {b \left (x-2 \sqrt {a} \left (c-b x^2\right )\right )}}{\sqrt {2} \sqrt {a^{3/2} b}} \\ \end{align*}