problem 976

Internal problem ID [9309]
Internal ﬁle name [OUTPUT/8245_Monday_June_06_2022_02_29_51_AM_52833150/index.tex]

Book: Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear ﬁrst order
Problem number: 976.
ODE order: 1.
ODE degree: 1.

\[ \boxed {y^{\prime }-\frac {y \left (y^{2} x^{7}+y x^{4}+x -3\right )}{x}=0} \]

2.399.1 Solving as abelFirstKind ode

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 }&=x^{6} y^{3}+y^{2} x^{3}+\frac {\left (x -3\right ) y}{x}\tag {1} \end {align*}

Therefore \begin {align*} f_0(x) &= 0\\ f_1(x) &= 1-\frac {3}{x}\\ f_2(x) &= x^{3}\\ f_3(x) &= x^{6} \end {align*}

Since $f_2(x)=x^{3}$ 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 {x^{3}}{3 x^{6}} \right ) \\ &= u \left (x \right )-\frac {1}{3 x^{3}} \end {align*}

The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = x^{6} u \left (x \right )^{3}+\frac {2 u \left (x \right )}{3}-\frac {3 u \left (x \right )}{x}-\frac {7}{27 x^{3}}\tag {2} \end {align*}

This is Abel ﬁrst kind ODE, it has the form \[ u^{\prime }\left (x \right )= f_0(x)+f_1(x) u \left (x \right ) +f_2(x)u \left (x \right )^{2}+f_3(x)u \left (x \right )^{3} \] Comparing the above to given ODE which is \begin {align*} u^{\prime }\left (x \right )&=x^{6} u \left (x \right )^{3}+\frac {\left (18 x^{3}-81 x^{2}\right ) u \left (x \right )}{27 x^{3}}-\frac {7}{27 x^{3}}\tag {1} \end {align*}

Therefore \begin {align*} f_0(x) &= -\frac {7}{27 x^{3}}\\ f_1(x) &= \frac {2}{3}-\frac {3}{x}\\ f_2(x) &= 0\\ f_3(x) &= x^{6} \end {align*}

Since $f_2(x)=0$ then we check the Abel invariant to see if it depends on $x$ or not. The Abel invariant is given by \begin {align*} -\frac {f_{1}^{3}}{f_{0}^{2} f_{3}} \end {align*}

Which when evaluating gives \begin {align*} \frac {531441 \left (-\frac {7 x^{2}}{3}-\frac {7 x^{3} \left (\frac {2}{3}-\frac {3}{x}\right )}{9}\right )^{3}}{16807 x^{9}} \end {align*}

Since the Abel invariant depends on $x$ then unable to solve this ode at this time.

2.399.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{3} x^{7}+y^{2} x^{4}+y x -y^{\prime } x -3 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y^{3} x^{7}-y^{2} x^{4}-y x +3 y}{x} \end {array} \]

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`

\[ y \left (x \right ) = \frac {\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (-\sqrt {3}\, \ln \left (3\right )-\sqrt {3}\, \ln \left (-\frac {1}{-2+\sqrt {3}\, \sin \left (2 \textit {\_Z} \right )+\cos \left (2 \textit {\_Z} \right )}\right )+\sqrt {3}\, \ln \left (7\right )+3 \sqrt {3}\, c_{1} -2 \sqrt {3}\, x -2 \textit {\_Z} \right )\right )-1}{2 x^{3}} \]

\[ \text {Solve}\left [-\frac {7}{3} \text {RootSum}\left [-7 \text {$\#$1}^3+6 \sqrt [3]{-7} \text {$\#$1}-7\&,\frac {\log \left (\frac {3 x^6 y(x)+x^3}{\sqrt [3]{7} \sqrt [3]{-x^9}}-\text {$\#$1}\right )}{2 \sqrt [3]{-7}-7 \text {$\#$1}^2}\&\right ]=\frac {7^{2/3} \left (-x^9\right )^{2/3}}{9 x^5}+c_1,y(x)\right ] \]

2.399 problem 976

2.399.1 Solving as abelFirstKind ode

2.399.2 Maple step by step solution