Internal problem ID [9990]
Internal file name [OUTPUT/8937_Monday_June_06_2022_06_00_15_AM_8884065/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1669 (book 6.78).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_integrable_as_is"
Maple gives the following as the ode type
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {x y^{\prime \prime }+\left (y-1\right ) y^{\prime }=0} \]
Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (x y^{\prime \prime }+\left (y-1\right ) y^{\prime }\right )d x &= 0 \\ \frac {y^{2}}{2}-2 y+x y^{\prime } = c_{1} \end {align*}
Which is now solved for \(y\). In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {-\frac {1}{2} y^{2}+2 y +c_{1}}{x} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(y)=-\frac {1}{2} y^{2}+2 y +c_{1}\). Integrating both sides gives \begin{align*} \frac {1}{-\frac {1}{2} y^{2}+2 y +c_{1}} \,dy &= \frac {1}{x} \,d x \\ \int { \frac {1}{-\frac {1}{2} y^{2}+2 y +c_{1}} \,dy} &= \int {\frac {1}{x} \,d x} \\ -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y +4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}&=\ln \left (x \right )+c_{2} \\ \end{align*} The solution is \[ -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y+4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}-\ln \left (x \right )-c_{2} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y+4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}-\ln \left (x \right )-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y+4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}-\ln \left (x \right )-c_{2} = 0 \] Verified OK.
Writing the ode as \[ x y^{\prime \prime }+\left (y-1\right ) y^{\prime } = 0 \] Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (x y^{\prime \prime }+\left (y-1\right ) y^{\prime }\right )d x &= 0 \\ \frac {y^{2}}{2}-2 y+x y^{\prime } = c_{1} \end {align*}
Which is now solved for \(y\). In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {-\frac {1}{2} y^{2}+2 y +c_{1}}{x} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(y)=-\frac {1}{2} y^{2}+2 y +c_{1}\). Integrating both sides gives \begin{align*} \frac {1}{-\frac {1}{2} y^{2}+2 y +c_{1}} \,dy &= \frac {1}{x} \,d x \\ \int { \frac {1}{-\frac {1}{2} y^{2}+2 y +c_{1}} \,dy} &= \int {\frac {1}{x} \,d x} \\ -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y +4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}&=\ln \left (x \right )+c_{2} \\ \end{align*} The solution is \[ -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y+4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}-\ln \left (x \right )-c_{2} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y+4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}-\ln \left (x \right )-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y+4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}-\ln \left (x \right )-c_{2} = 0 \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases trying symmetries linear in x and y(x) `, `-> Computing symmetries using: way = 3 Try integration with the canonical coordinates of the symmetry [x, 0] -> Calling odsolve with the ODE`, diff(_b(_a), _a) = _a*_b(_a)^2-2*_b(_a)^2, _b(_a), explicit, HINT = [[-_a+2, 2*_b]]` *** Subleve symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[-_a+2, 2*_b]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 24
dsolve(x*diff(diff(y(x),x),x)+(y(x)-1)*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 c_{1} +\tanh \left (\frac {\ln \left (x \right )-c_{2}}{2 c_{1}}\right )}{c_{1}} \]
✓ Solution by Mathematica
Time used: 60.069 (sec). Leaf size: 46
DSolve[(-1 + y[x])*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2-\sqrt {2} \sqrt {2+c_1} \tanh \left (\frac {\sqrt {2+c_1} (-\log (x)+2 c_2)}{\sqrt {2}}\right ) \]