Internal problem ID [9232]
Internal file name [OUTPUT/8168_Monday_June_06_2022_02_00_43_AM_79190799/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 899.
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, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]
\[ \boxed {y^{\prime }-\frac {32 x^{5}+64 x^{6}+64 y^{2} x^{6}+32 y x^{4}+4 x^{2}+64 x^{6} y^{3}+48 y^{2} x^{4}+12 x^{2} y+1}{64 x^{8}}=0} \]
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^{2}}+\frac {\left (64 x^{6}+48 x^{4}\right ) y^{2}}{64 x^{8}}+\frac {\left (32 x^{4}+12 x^{2}\right ) y}{64 x^{8}}+\frac {64 x^{6}+32 x^{5}+4 x^{2}+1}{64 x^{8}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= \frac {1}{x^{2}}+\frac {1}{2 x^{3}}+\frac {1}{16 x^{6}}+\frac {1}{64 x^{8}}\\ f_1(x) &= \frac {1}{2 x^{4}}+\frac {3}{16 x^{6}}\\ f_2(x) &= \frac {1}{x^{2}}+\frac {3}{4 x^{4}}\\ f_3(x) &= \frac {1}{x^{2}} \end {align*}
Since \(f_2(x)=\frac {1}{x^{2}}+\frac {3}{4 x^{4}}\) 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 {1}{x^{2}}+\frac {3}{4 x^{4}}}{\frac {3}{x^{2}}} \right ) \\ &= u \left (x \right )-\frac {4 x^{2}+3}{12 x^{2}} \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^{2}}-\frac {u \left (x \right )}{3 x^{2}}+\frac {29}{27 x^{2}}\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^{2}} \end {align*}
Where \(f(x)=\frac {1}{x^{2}}\) 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^{2}} \,d x \\ \int { \frac {1}{u^{3}-\frac {1}{3} u +\frac {29}{27}} \,du} &= \int {\frac {1}{x^{2}} \,d x} \\ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}&=-\frac {1}{x}+c_{2} \\ \end{align*} Which results in \[ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}=-\frac {1}{x}+c_{2} \] The solution is \[ \int _{}^{u \left (x \right )}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} +\frac {1}{x}-c_{2} = 0 \] Substituting \(u=y-\frac {\left (\frac {1}{x^{2}}+\frac {3}{4 x^{4}}\right ) x^{2}}{3}\) in the above solution gives \begin {align*} \int _{}^{y-\frac {\left (\frac {1}{x^{2}}+\frac {3}{4 x^{4}}\right ) x^{2}}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} +\frac {1}{x}-c_{2} = 0 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y-\frac {\left (\frac {1}{x^{2}}+\frac {3}{4 x^{4}}\right ) x^{2}}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} +\frac {1}{x}-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ \int _{}^{y-\frac {\left (\frac {1}{x^{2}}+\frac {3}{4 x^{4}}\right ) x^{2}}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} +\frac {1}{x}-c_{2} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 64 y^{\prime } x^{8}-64 x^{6} y^{3}-64 y^{2} x^{6}-48 y^{2} x^{4}-64 x^{6}-32 y x^{4}-32 x^{5}-12 x^{2} y-4 x^{2}-1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {32 x^{5}+64 x^{6}+64 y^{2} x^{6}+32 y x^{4}+4 x^{2}+64 x^{6} y^{3}+48 y^{2} x^{4}+12 x^{2} y+1}{64 x^{8}} \end {array} \]
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: 47
dsolve(diff(y(x),x) = 1/64*(32*x^5+64*x^6+64*y(x)^2*x^6+32*y(x)*x^4+4*x^2+64*x^6*y(x)^3+48*x^4*y(x)^2+12*x^2*y(x)+1)/x^8,y(x), singsol=all)
\[ y \left (x \right ) = \frac {116 \operatorname {RootOf}\left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right ) x +3 c_{1} x -1\right ) x^{2}-12 x^{2}-9}{36 x^{2}} \]
✓ Solution by Mathematica
Time used: 0.164 (sec). Leaf size: 106
DSolve[y'[x] == (1/64 + x^2/16 + x^5/2 + x^6 + (3*x^2*y[x])/16 + (x^4*y[x])/2 + (3*x^4*y[x]^2)/4 + x^6*y[x]^2 + x^6*y[x]^3)/x^8,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {\frac {3 y(x)}{x^2}+\frac {4 x^2+3}{4 x^4}}{\sqrt [3]{29} \sqrt [3]{\frac {1}{x^6}}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=-\frac {1}{9} 29^{2/3} \left (\frac {1}{x^6}\right )^{2/3} x^3+c_1,y(x)\right ] \]