Internal problem ID [9288]
Internal file name [OUTPUT/8224_Monday_June_06_2022_02_21_17_AM_23518876/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 955.
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 C`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {-150 x^{3} y+60 x^{6}+350 x^{\frac {7}{2}}-150 x^{3}-125 y \sqrt {x}+250 x -125 \sqrt {x}-125 y^{3}+150 x^{3} y^{2}+750 y^{2} \sqrt {x}-60 y x^{6}-600 x^{\frac {7}{2}} y-1500 x y+8 x^{9}+120 x^{\frac {13}{2}}+600 x^{4}+1000 x^{\frac {3}{2}}}{25 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -120 x^{\frac {13}{2}}-8 x^{9}+600 x^{\frac {7}{2}} y+60 y x^{6}-350 x^{\frac {7}{2}}-60 x^{6}-150 x^{3} y^{2}+50 y^{\prime } x^{4}+150 x^{3} y-600 x^{4}+125 y^{3}-750 y^{2} \sqrt {x}-125 y^{\prime } y x +250 y^{\prime } x^{\frac {3}{2}}-1000 x^{\frac {3}{2}}+150 x^{3}+125 y \sqrt {x}+1500 x y-125 y^{\prime } x +125 \sqrt {x}-250 x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-150 x^{3} y+60 x^{6}+350 x^{\frac {7}{2}}-150 x^{3}-125 y \sqrt {x}+250 x -125 \sqrt {x}-125 y^{3}+150 x^{3} y^{2}+750 y^{2} \sqrt {x}-60 y x^{6}-600 x^{\frac {7}{2}} y-1500 x y+8 x^{9}+120 x^{\frac {13}{2}}+600 x^{4}+1000 x^{\frac {3}{2}}}{50 x^{4}-125 x y+250 x^{\frac {3}{2}}-125 x} \end {array} \]
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: 0.016 (sec). Leaf size: 101
dsolve(diff(y(x),x) = 1/25*(-150*x^3*y(x)+60*x^6+350*x^(7/2)-150*x^3-125*y(x)*x^(1/2)+250*x-125*x^(1/2)-125*y(x)^3+150*x^3*y(x)^2+750*y(x)^2*x^(1/2)-60*y(x)*x^6-600*y(x)*x^(7/2)-1500*x*y(x)+8*x^9+120*x^(13/2)+600*x^4+1000*x^(3/2))/(-5*y(x)+2*x^3+10*x^(1/2)-5)/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (2 x^{3}+10 \sqrt {x}\right ) \sqrt {c_{1} -2 \ln \left (x \right )}-2 x^{3}-10 \sqrt {x}+5}{5 \sqrt {c_{1} -2 \ln \left (x \right )}-5} \\ y \left (x \right ) &= \frac {\left (2 x^{3}+10 \sqrt {x}\right ) \sqrt {c_{1} -2 \ln \left (x \right )}+2 x^{3}+10 \sqrt {x}-5}{5 \sqrt {c_{1} -2 \ln \left (x \right )}+5} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.647 (sec). Leaf size: 92
DSolve[y'[x] == (-5*Sqrt[x] + 10*x + 40*x^(3/2) - 6*x^3 + 14*x^(7/2) + 24*x^4 + (12*x^6)/5 + (24*x^(13/2))/5 + (8*x^9)/25 - 5*Sqrt[x]*y[x] - 60*x*y[x] - 6*x^3*y[x] - 24*x^(7/2)*y[x] - (12*x^6*y[x])/5 + 30*Sqrt[x]*y[x]^2 + 6*x^3*y[x]^2 - 5*y[x]^3)/(x*(-5 + 10*Sqrt[x] + 2*x^3 - 5*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 x^3}{5}+2 \sqrt {x}-\frac {125}{125+\sqrt {-31250 \log (x)+c_1}} \\ y(x)\to \frac {2 x^3}{5}+2 \sqrt {x}+\frac {125}{-125+\sqrt {-31250 \log (x)+c_1}} \\ y(x)\to \frac {2}{5} \left (x^3+5 \sqrt {x}\right ) \\ \end{align*}