Internal problem ID [12269]
Internal file name [OUTPUT/10922_Thursday_September_28_2023_01_08_53_AM_71924558/index.tex]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page
221
Problem number: Problem 19(b).
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]]
\[ \boxed {\frac {x y^{\prime \prime }}{y+1}+\frac {y y^{\prime }-x {y^{\prime }}^{2}+y^{\prime }}{\left (y+1\right )^{2}}=\sin \left (x \right ) x} \]
Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (\frac {x y^{\prime \prime }}{y+1}+\left (\frac {y}{\left (y+1\right )^{2}}-\frac {x y^{\prime }}{\left (y+1\right )^{2}}+\frac {1}{\left (y+1\right )^{2}}\right ) y^{\prime }\right )d x &= \int \sin \left (x \right ) x d x\\ \left (\frac {x y}{\left (y+1\right )^{2}}+\frac {x}{\left (y+1\right )^{2}}\right ) y^{\prime } = \sin \left (x \right )-\cos \left (x \right ) x + 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 {\left (\sin \left (x \right )-\cos \left (x \right ) x +c_{1} \right ) \left (1+y \right )}{x} \end {align*}
Where \(f(x)=\frac {\sin \left (x \right )-\cos \left (x \right ) x +c_{1}}{x}\) and \(g(y)=1+y\). Integrating both sides gives \begin{align*} \frac {1}{1+y} \,dy &= \frac {\sin \left (x \right )-\cos \left (x \right ) x +c_{1}}{x} \,d x \\ \int { \frac {1}{1+y} \,dy} &= \int {\frac {\sin \left (x \right )-\cos \left (x \right ) x +c_{1}}{x} \,d x} \\ \ln \left (1+y \right )&=-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )+c_{2} \\ \end{align*} Raising both side to exponential gives \begin {align*} 1+y &= {\mathrm e}^{-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )+c_{2}} \end {align*}
Which simplifies to \begin {align*} 1+y &= c_{3} {\mathrm e}^{-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{3} {\mathrm e}^{-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )+c_{2}}-1 \\ \end{align*}
Verification of solutions
\[ y = c_{3} {\mathrm e}^{-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )+c_{2}}-1 \] Verified OK.
Writing the ode as \[ \frac {x y^{\prime \prime }}{y+1}+\left (\frac {y}{\left (y+1\right )^{2}}-\frac {x y^{\prime }}{\left (y+1\right )^{2}}+\frac {1}{\left (y+1\right )^{2}}\right ) y^{\prime } = \sin \left (x \right ) x \] Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (\frac {x y^{\prime \prime }}{y+1}+\left (\frac {y}{\left (y+1\right )^{2}}-\frac {x y^{\prime }}{\left (y+1\right )^{2}}+\frac {1}{\left (y+1\right )^{2}}\right ) y^{\prime }\right )d x &= \int \sin \left (x \right ) x d x\\ \frac {x y^{\prime }}{y+1} = \sin \left (x \right )-\cos \left (x \right ) x +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 {\left (\sin \left (x \right )-\cos \left (x \right ) x +c_{1} \right ) \left (1+y \right )}{x} \end {align*}
Where \(f(x)=\frac {\sin \left (x \right )-\cos \left (x \right ) x +c_{1}}{x}\) and \(g(y)=1+y\). Integrating both sides gives \begin{align*} \frac {1}{1+y} \,dy &= \frac {\sin \left (x \right )-\cos \left (x \right ) x +c_{1}}{x} \,d x \\ \int { \frac {1}{1+y} \,dy} &= \int {\frac {\sin \left (x \right )-\cos \left (x \right ) x +c_{1}}{x} \,d x} \\ \ln \left (1+y \right )&=-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )+c_{2} \\ \end{align*} Raising both side to exponential gives \begin {align*} 1+y &= {\mathrm e}^{-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )+c_{2}} \end {align*}
Which simplifies to \begin {align*} 1+y &= c_{3} {\mathrm e}^{-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{3} {\mathrm e}^{-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )+c_{2}}-1 \\ \end{align*}
Verification of solutions
\[ y = c_{3} {\mathrm e}^{-\sin \left (x \right )+\operatorname {Si}\left (x \right )+c_{1} \ln \left (x \right )+c_{2}}-1 \] 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 a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type <- LODE of Euler type successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful`
✓ Solution by Maple
Time used: 0.219 (sec). Leaf size: 26
dsolve(x*diff(y(x),x$2)/(1+y(x))+( y(x)*diff(y(x),x)-x* diff(y(x),x)^2+diff(y(x),x))/( 1+y(x))^2=x*sin(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-c_{2}} {\mathrm e}^{\operatorname {Si}\left (x \right )-\sin \left (x \right )-\frac {\pi \,\operatorname {csgn}\left (x \right )}{2}}-1 \]
✓ Solution by Mathematica
Time used: 1.681 (sec). Leaf size: 28
DSolve[x*y''[x]/(1+y[x])+( y[x]*y'[x]-x* y'[x]^2+y'[x])/( 1+y[x])^2==x*Sin[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -1+x^{c_2} e^{\text {Si}(x)-\sin (x)+c_1} \\ y(x)\to -1 \\ \end{align*}