Internal problem ID [9067]
Internal file name [OUTPUT/8002_Monday_June_06_2022_01_10_28_AM_82642248/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 733.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {2 x \sin \left (x \right )-\ln \left (2 x \right )+\ln \left (2 x \right ) x^{4}-2 \ln \left (2 x \right ) x^{2} y+\ln \left (2 x \right ) y^{2}}{\sin \left (x \right )}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {2 x \sin \left (x \right )-\ln \left (2 x \right )+\ln \left (2 x \right ) x^{4}-2 \ln \left (2 x \right ) x^{2} y +\ln \left (2 x \right ) y^{2}}{\sin \left (x \right )} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = 2 x -\frac {\ln \left (2\right )}{\sin \left (x \right )}-\frac {\ln \left (x \right )}{\sin \left (x \right )}+\frac {\ln \left (2\right ) x^{4}}{\sin \left (x \right )}+\frac {x^{4} \ln \left (x \right )}{\sin \left (x \right )}-\frac {2 \ln \left (2\right ) x^{2} y}{\sin \left (x \right )}-\frac {2 y \,x^{2} \ln \left (x \right )}{\sin \left (x \right )}+\frac {y^{2} \ln \left (2\right )}{\sin \left (x \right )}+\frac {y^{2} \ln \left (x \right )}{\sin \left (x \right )} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {\ln \left (2 x \right ) x^{4}+2 x \sin \left (x \right )-\ln \left (2 x \right )}{\sin \left (x \right )}\), \(f_1(x)=-\frac {2 \ln \left (2 x \right ) x^{2}}{\sin \left (x \right )}\) and \(f_2(x)=\frac {\ln \left (2 x \right )}{\sin \left (x \right )}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {\ln \left (2 x \right ) u}{\sin \left (x \right )}} \tag {1} \end {align*}
Using the above substitution in the given ODE results (after some simplification)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=-\frac {\ln \left (2 x \right ) \cos \left (x \right )}{\sin \left (x \right )^{2}}+\frac {1}{x \sin \left (x \right )}\\ f_1 f_2 &=-\frac {2 \ln \left (2 x \right )^{2} x^{2}}{\sin \left (x \right )^{2}}\\ f_2^2 f_0 &=\frac {\ln \left (2 x \right )^{2} \left (\ln \left (2 x \right ) x^{4}+2 x \sin \left (x \right )-\ln \left (2 x \right )\right )}{\sin \left (x \right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {\ln \left (2 x \right ) u^{\prime \prime }\left (x \right )}{\sin \left (x \right )}-\left (-\frac {\ln \left (2 x \right ) \cos \left (x \right )}{\sin \left (x \right )^{2}}+\frac {1}{x \sin \left (x \right )}-\frac {2 \ln \left (2 x \right )^{2} x^{2}}{\sin \left (x \right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {\ln \left (2 x \right )^{2} \left (\ln \left (2 x \right ) x^{4}+2 x \sin \left (x \right )-\ln \left (2 x \right )\right ) u \left (x \right )}{\sin \left (x \right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives Unable to solve. Terminating.
✗ Solution by Maple
dsolve(diff(y(x),x) = (2*x*sin(x)-ln(2*x)+ln(2*x)*x^4-2*ln(2*x)*x^2*y(x)+ln(2*x)*y(x)^2)/sin(x),y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 17.66 (sec). Leaf size: 82
DSolve[y'[x] == Csc[x]*(-Log[2*x] + x^4*Log[2*x] + 2*x*Sin[x] - 2*x^2*Log[2*x]*y[x] + Log[2*x]*y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\exp \left (\int _1^x2 \csc (K[5]) \log (2 K[5])dK[5]\right )}{-\int _1^x\exp \left (\int _1^{K[6]}2 \csc (K[5]) \log (2 K[5])dK[5]\right ) \csc (K[6]) \log (2 K[6])dK[6]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ \end{align*}