67.2.49 problem Problem 19(b)
Internal
problem
ID
[14006]
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)
Date
solved
:
Tuesday, January 28, 2025 at 06:11:39 AM
CAS
classification
:
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} \frac {x y^{\prime \prime }}{y+1}+\frac {y^{\prime } y-x {y^{\prime }}^{2}+y^{\prime }}{\left (y+1\right )^{2}}&=x \sin \left (x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.539 (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)
\[
y = c_{1} x^{-c_{2}} {\mathrm e}^{\operatorname {Si}\left (x \right )-\sin \left (x \right )-\frac {\pi \,\operatorname {csgn}\left (x \right )}{2}}-1
\]
✓ Solution by Mathematica
Time used: 60.330 (sec). Leaf size: 450
DSolve[x*D[y[x],{x,2}]/(1+y[x])+( y[x]*D[y[x],x]-x* D[y[x],x]^2+D[y[x],x])/( 1+y[x])^2==x*Sin[x],y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to -\frac {x \log (x) \exp \left (c_2-\int _1^x\frac {c_1{}^2+\log (K[3]) c_1+c_1+\int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]{}^2+K[3]^2 \log ^2(K[3]) \sin (K[3])+(2 c_1+\log (K[3])+1) \int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]+y(K[3]) \left (K[3]^2 \sin (K[3]) \log ^2(K[3])+\int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]{}^2+c_1 (c_1+\log (K[3])+1)+(2 c_1+\log (K[3])+1) \int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]\right )+K[3] \log (K[3]) y''(K[3])}{K[3] \log (K[3]) (y(K[3])+1) \left (c_1+\int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]\right )}dK[3]\right )+\int _1^x-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]+c_1}{\int _1^x-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]+c_1}
\]