18.2.15 problem Problem 15.33

Internal problem ID [3498]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 15, Higher order ordinary differential equations. 15.4 Exercises, page 523
Problem number : Problem 15.33
Date solved : Tuesday, January 28, 2025 at 02:39:17 PM
CAS classification : [[_3rd_order, _exact, _nonlinear]]

\begin{align*} 2 y y^{\prime \prime \prime }+2 \left (y+3 y^{\prime }\right ) y^{\prime \prime }+2 {y^{\prime }}^{2}&=\sin \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 81

dsolve(2*y(x)*diff(y(x),x$3)+2*(y(x)+3*diff(y(x),x))*diff(y(x),x$2)+2*(diff(y(x),x))^2=sin(x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-4 \,{\mathrm e}^{x} \left (\left (-\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}+c_{1} \left (x -1\right )+c_3 \right ) {\mathrm e}^{x}-c_{2} \right )}\, {\mathrm e}^{-x}}{2} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-4 \,{\mathrm e}^{x} \left (\left (-\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}+c_{1} \left (x -1\right )+c_3 \right ) {\mathrm e}^{x}-c_{2} \right )}\, {\mathrm e}^{-x}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.458 (sec). Leaf size: 88

DSolve[2*y[x]*D[y[x],{x,3}]+2*(y[x]+3*D[y[x],x])*D[y[x],{x,2}]+2*(D[y[x],x])^2==Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {-\sin (x)+\cos (x)+2 c_1 x+2 c_3 e^{-x}-2 c_1-4 c_2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-\sin (x)+\cos (x)+2 c_1 x+2 c_3 e^{-x}-2 c_1-4 c_2}}{\sqrt {2}} \\ \end{align*}