Internal problem ID [9291]
Internal file name [OUTPUT/8227_Monday_June_06_2022_02_22_54_AM_22411005/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 958.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]
\[ \boxed {y^{\prime }-\frac {2 x +4 y \ln \left (2 x +1\right ) x +6 y^{2} \ln \left (2 x +1\right ) x +6 y \ln \left (2 x +1\right )^{2} x +2 \ln \left (2 x +1\right )^{3} x +2 x y^{3}+2 \ln \left (2 x +1\right )^{2} x +2 y^{2} x -1+3 y^{2} \ln \left (2 x +1\right )+3 y \ln \left (2 x +1\right )^{2}+y^{2}+y^{3}+2 y \ln \left (2 x +1\right )+\ln \left (2 x +1\right )^{2}+\ln \left (2 x +1\right )^{3}}{2 x +1}=0} \]
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 }&=y^{3}+\frac {\left (6 \ln \left (2 x +1\right ) x +3 \ln \left (2 x +1\right )+2 x +1\right ) y^{2}}{2 x +1}+\frac {\left (6 \ln \left (2 x +1\right )^{2} x +3 \ln \left (2 x +1\right )^{2}+4 \ln \left (2 x +1\right ) x +2 \ln \left (2 x +1\right )\right ) y}{2 x +1}+\frac {2 \ln \left (2 x +1\right )^{3} x +\ln \left (2 x +1\right )^{3}+2 \ln \left (2 x +1\right )^{2} x +\ln \left (2 x +1\right )^{2}+2 x -1}{2 x +1}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= \frac {2 \ln \left (2 x +1\right )^{3} x}{2 x +1}+\frac {\ln \left (2 x +1\right )^{3}}{2 x +1}+\frac {2 \ln \left (2 x +1\right )^{2} x}{2 x +1}+\frac {\ln \left (2 x +1\right )^{2}}{2 x +1}+\frac {2 x}{2 x +1}-\frac {1}{2 x +1}\\ f_1(x) &= \frac {6 \ln \left (2 x +1\right )^{2} x}{2 x +1}+\frac {3 \ln \left (2 x +1\right )^{2}}{2 x +1}+\frac {4 \ln \left (2 x +1\right ) x}{2 x +1}+\frac {2 \ln \left (2 x +1\right )}{2 x +1}\\ f_2(x) &= \frac {6 \ln \left (2 x +1\right ) x}{2 x +1}+\frac {3 \ln \left (2 x +1\right )}{2 x +1}+\frac {2 x}{2 x +1}+\frac {1}{2 x +1}\\ f_3(x) &= 1 \end {align*}
Since \(f_2(x)=\frac {6 \ln \left (2 x +1\right ) x}{2 x +1}+\frac {3 \ln \left (2 x +1\right )}{2 x +1}+\frac {2 x}{2 x +1}+\frac {1}{2 x +1}\) 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 \ln \left (2 x +1\right ) x}{2 x +1}+\frac {3 \ln \left (2 x +1\right )}{2 x +1}+\frac {2 x}{2 x +1}+\frac {1}{2 x +1}}{3} \right ) \\ &= -\ln \left (2 x +1\right )+u \left (x \right )-\frac {1}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = u \left (x \right )^{3}-\frac {u \left (x \right )}{3}+\frac {29}{27}\tag {2} \end {align*}
The above ODE (2) can now be solved as separable.
Integrating both sides gives \begin {align*} \int _{}^{u \left (x \right )}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \end {align*}
Substituting \(u=y-\frac {2 \ln \left (2 x +1\right ) x}{2 x +1}-\frac {\ln \left (2 x +1\right )}{2 x +1}-\frac {2 x}{3 \left (2 x +1\right )}-\frac {1}{3 \left (2 x +1\right )}\) in the above solution gives \begin {align*} \int _{}^{y-\frac {2 \ln \left (2 x +1\right ) x}{2 x +1}-\frac {\ln \left (2 x +1\right )}{2 x +1}-\frac {2 x}{3 \left (2 x +1\right )}-\frac {1}{3 \left (2 x +1\right )}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y-\frac {2 \ln \left (2 x +1\right ) x}{2 x +1}-\frac {\ln \left (2 x +1\right )}{2 x +1}-\frac {2 x}{3 \left (2 x +1\right )}-\frac {1}{3 \left (2 x +1\right )}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y-\frac {2 \ln \left (2 x +1\right ) x}{2 x +1}-\frac {\ln \left (2 x +1\right )}{2 x +1}-\frac {2 x}{3 \left (2 x +1\right )}-\frac {1}{3 \left (2 x +1\right )}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x y^{3}+6 y^{2} \ln \left (2 x +1\right ) x +6 y \ln \left (2 x +1\right )^{2} x +2 \ln \left (2 x +1\right )^{3} x +y^{3}+3 y^{2} \ln \left (2 x +1\right )+2 y^{2} x +3 y \ln \left (2 x +1\right )^{2}+4 y \ln \left (2 x +1\right ) x +\ln \left (2 x +1\right )^{3}+2 \ln \left (2 x +1\right )^{2} x +y^{2}+2 y \ln \left (2 x +1\right )+\ln \left (2 x +1\right )^{2}-2 y^{\prime } x -y^{\prime }+2 x -1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-2 x -4 y \ln \left (2 x +1\right ) x -6 y^{2} \ln \left (2 x +1\right ) x -6 y \ln \left (2 x +1\right )^{2} x -2 \ln \left (2 x +1\right )^{3} x -2 x y^{3}-2 \ln \left (2 x +1\right )^{2} x -2 y^{2} x +1-3 y^{2} \ln \left (2 x +1\right )-3 y \ln \left (2 x +1\right )^{2}-y^{2}-y^{3}-2 y \ln \left (2 x +1\right )-\ln \left (2 x +1\right )^{2}-\ln \left (2 x +1\right )^{3}}{-2 x -1} \end {array} \]
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: 40
dsolve(diff(y(x),x) = 1/(2*x+1)*(2*x+4*y(x)*ln(2*x+1)*x+6*y(x)^2*ln(2*x+1)*x+6*y(x)*ln(2*x+1)^2*x+2*ln(2*x+1)^3*x+2*x*y(x)^3+2*ln(2*x+1)^2*x+2*x*y(x)^2-1+3*y(x)^2*ln(2*x+1)+3*y(x)*ln(2*x+1)^2+y(x)^2+y(x)^3+2*y(x)*ln(2*x+1)+ln(2*x+1)^2+ln(2*x+1)^3),y(x), singsol=all)
\[ y \left (x \right ) = -\ln \left (2 x +1\right )-\frac {1}{3}+\frac {29 \operatorname {RootOf}\left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+x +3 c_{1} \right )}{9} \]
✓ Solution by Mathematica
Time used: 0.278 (sec). Leaf size: 82
DSolve[y'[x] == (-1 + 2*x + Log[1 + 2*x]^2 + 2*x*Log[1 + 2*x]^2 + Log[1 + 2*x]^3 + 2*x*Log[1 + 2*x]^3 + 2*Log[1 + 2*x]*y[x] + 4*x*Log[1 + 2*x]*y[x] + 3*Log[1 + 2*x]^2*y[x] + 6*x*Log[1 + 2*x]^2*y[x] + y[x]^2 + 2*x*y[x]^2 + 3*Log[1 + 2*x]*y[x]^2 + 6*x*Log[1 + 2*x]*y[x]^2 + y[x]^3 + 2*x*y[x]^3)/(1 + 2*x),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {3 y(x)+3 \log (2 x+1)+1}{\sqrt [3]{29}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=\frac {1}{9} 29^{2/3} x+c_1,y(x)\right ] \]