Internal problem ID [3840]
Internal file name [OUTPUT/3333_Sunday_June_05_2022_09_09_26_AM_19785721/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 21
Problem number: 590.
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, [_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {x^{7} y y^{\prime }-5 x^{3} y=2 x^{2}+2} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{7} y y^{\prime }-5 x^{3} y=2 x^{2}+2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2 x^{2}+2+5 x^{3} y}{x^{7} y} \end {array} \]
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 trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 120
dsolve(x^7*y(x)*diff(y(x),x) = 2*x^2+2+5*x^3*y(x),y(x), singsol=all)
\[ -\frac {\left (y \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x^{3} y \left (x \right )+1\right )^{2}}{x^{2}}\right ) x^{3}-c_{1} x +\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x^{3} y \left (x \right )+1\right )^{2}}{x^{2}}\right )\right ) \left (\frac {x^{6} y \left (x \right )^{2}+2 x^{3} y \left (x \right )+x^{2}+1}{x^{2}}\right )^{\frac {1}{4}}+2 x^{2}}{\left (\frac {x^{6} y \left (x \right )^{2}+2 x^{3} y \left (x \right )+x^{2}+1}{x^{2}}\right )^{\frac {1}{4}} x} = 0 \]
✓ Solution by Mathematica
Time used: 0.381 (sec). Leaf size: 98
DSolve[x^7 y[x] y'[x]==2(1+x^2)+5 x^3 y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\frac {\frac {i \left (x^3 y(x)+1\right ) \sqrt [4]{x^4 y(x)^2+\frac {1}{x^2}+2 x y(x)+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},-\frac {\left (y(x) x^3+1\right )^2}{x^2}\right )}{2 x}+i x}{\sqrt [4]{-\frac {\left (x^3 y(x)+1\right )^2}{x^2}-1}},y(x)\right ] \]