Internal problem ID [9287]
Internal file name [OUTPUT/8223_Monday_June_06_2022_02_21_01_AM_78295517/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 954.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[_rational, _Abel]
\[ \boxed {y^{\prime }-\frac {150 x^{3}+125 \sqrt {x}+125+125 y^{2}-100 x^{3} y-500 y \sqrt {x}+20 x^{6}+200 x^{\frac {7}{2}}+500 x +125 y^{3}-150 y^{2} x^{3}-750 y^{2} \sqrt {x}+60 y x^{6}+600 y x^{\frac {7}{2}}+1500 x y-8 x^{9}-120 x^{\frac {13}{2}}-600 x^{4}-1000 x^{\frac {3}{2}}}{125 x}=0} \]
This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(x)+f_1(x) y +f_2(x)y^{2}+f_3(x)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=\frac {y^{3}}{x}-\frac {\left (150 x^{3}+750 \sqrt {x}-125\right ) y^{2}}{125 x}-\frac {\left (-600 x^{\frac {7}{2}}-60 x^{6}+100 x^{3}+500 \sqrt {x}-1500 x \right ) y}{125 x}-\frac {120 x^{\frac {13}{2}}+8 x^{9}-200 x^{\frac {7}{2}}-20 x^{6}+600 x^{4}+1000 x^{\frac {3}{2}}-150 x^{3}-125 \sqrt {x}-500 x -125}{125 x}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= -\frac {24 x^{\frac {11}{2}}}{25}-\frac {8 x^{8}}{125}+\frac {8 x^{\frac {5}{2}}}{5}+\frac {4 x^{5}}{25}-\frac {24 x^{3}}{5}-8 \sqrt {x}+\frac {6 x^{2}}{5}+\frac {1}{\sqrt {x}}+4+\frac {1}{x}\\ f_1(x) &= \frac {24 x^{\frac {5}{2}}}{5}+\frac {12 x^{5}}{25}-\frac {4 x^{2}}{5}-\frac {4}{\sqrt {x}}+12\\ f_2(x) &= -\frac {6 x^{2}}{5}-\frac {6}{\sqrt {x}}+\frac {1}{x}\\ f_3(x) &= \frac {1}{x} \end {align*}
Since \(f_2(x)=-\frac {6 x^{2}}{5}-\frac {6}{\sqrt {x}}+\frac {1}{x}\) is not zero, then the first step is to apply the following transformation to remove \(f_2\). Let \(y = u(x) - \frac {f_2}{3 f_3}\) or \begin {align*} y &= u(x) - \left ( \frac {-\frac {6 x^{2}}{5}-\frac {6}{\sqrt {x}}+\frac {1}{x}}{\frac {3}{x}} \right ) \\ &= u \left (x \right )-\frac {\left (-\frac {6 x^{2}}{5}-\frac {6}{\sqrt {x}}+\frac {1}{x}\right ) x}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \frac {u \left (x \right )^{3}}{x}-\frac {u \left (x \right )}{3 x}+\frac {29}{27 x}\tag {2} \end {align*}
The above ODE (2) can now be solved as separable.
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= \frac {u^{3}-\frac {1}{3} u +\frac {29}{27}}{x} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(u)=u^{3}-\frac {1}{3} u +\frac {29}{27}\). Integrating both sides gives \begin{align*} \frac {1}{u^{3}-\frac {1}{3} u +\frac {29}{27}} \,du &= \frac {1}{x} \,d x \\ \int { \frac {1}{u^{3}-\frac {1}{3} u +\frac {29}{27}} \,du} &= \int {\frac {1}{x} \,d x} \\ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}&=\ln \left (x \right )+c_{2} \\ \end{align*} Which results in \[ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}=\ln \left (x \right )+c_{2} \] The solution is \[ \int _{}^{u \left (x \right )}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} = 0 \] Substituting \(u=y-\frac {\left (-\frac {6 x^{2}}{5}-\frac {6}{\sqrt {x}}+\frac {1}{x}\right ) x}{3}\) in the above solution gives \begin {align*} \int _{}^{y-\frac {\left (-\frac {6 x^{2}}{5}-\frac {6}{\sqrt {x}}+\frac {1}{x}\right ) x}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} = 0 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y-\frac {\left (-\frac {6 x^{2}}{5}-\frac {6}{\sqrt {x}}+\frac {1}{x}\right ) x}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ \int _{}^{y-\frac {\left (-\frac {6 x^{2}}{5}-\frac {6}{\sqrt {x}}+\frac {1}{x}\right ) x}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 120 x^{\frac {13}{2}}+8 x^{9}-600 y x^{\frac {7}{2}}-200 x^{\frac {7}{2}}-60 y x^{6}-20 x^{6}+150 y^{2} x^{3}+100 x^{3} y+600 x^{4}+1000 x^{\frac {3}{2}}+750 y^{2} \sqrt {x}-125 y^{3}-150 x^{3}+125 y^{\prime } x +500 y \sqrt {x}-125 y^{2}-1500 x y-125 \sqrt {x}-500 x -125=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {150 x^{3}+125 \sqrt {x}+125+125 y^{2}-100 x^{3} y-500 y \sqrt {x}+20 x^{6}+200 x^{\frac {7}{2}}+500 x +125 y^{3}-150 y^{2} x^{3}-750 y^{2} \sqrt {x}+60 y x^{6}+600 y x^{\frac {7}{2}}+1500 x y-8 x^{9}-120 x^{\frac {13}{2}}-600 x^{4}-1000 x^{\frac {3}{2}}}{125 x} \end {array} \]
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: 0.0 (sec). Leaf size: 53
dsolve(diff(y(x),x) = 1/125*(150*x^3+125*x^(1/2)+125+125*y(x)^2-100*x^3*y(x)-500*y(x)*x^(1/2)+20*x^6+200*x^(7/2)+500*x+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))/x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {18 x^{\frac {7}{2}}+145 \operatorname {RootOf}\left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+\ln \left (x \right )+3 c_{1} \right ) \sqrt {x}-15 \sqrt {x}+90 x}{45 \sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.254 (sec). Leaf size: 115
DSolve[y'[x] == (1 + Sqrt[x] + 4*x - 8*x^(3/2) + (6*x^3)/5 + (8*x^(7/2))/5 - (24*x^4)/5 + (4*x^6)/25 - (24*x^(13/2))/25 - (8*x^9)/125 - 4*Sqrt[x]*y[x] + 12*x*y[x] - (4*x^3*y[x])/5 + (24*x^(7/2)*y[x])/5 + (12*x^6*y[x])/25 + y[x]^2 - 6*Sqrt[x]*y[x]^2 - (6*x^3*y[x]^2)/5 + y[x]^3)/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {\frac {-6 x^3-30 \sqrt {x}+5}{5 x}+\frac {3 y(x)}{x}}{\sqrt [3]{29} \sqrt [3]{\frac {1}{x^3}}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=\frac {1}{9} 29^{2/3} \left (\frac {1}{x^3}\right )^{2/3} x^2 \log (x)+c_1,y(x)\right ] \]