71.9.2 problem 2

Internal problem ID [14481]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.1, page 186
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 06:42:09 AM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} x y^{\prime \prime \prime }+x y^{\prime }&=4 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=1\\ y^{\prime \prime }\left (1\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.230 (sec). Leaf size: 44

dsolve([x*diff(y(x),x$3)+x*diff(y(x),x)=4,y(1) = 0, D(y)(1) = 1, (D@@2)(y)(1) = -1],y(x), singsol=all)
 

\[ y = \left (4 \,\operatorname {Ci}\left (1\right )-4 \,\operatorname {Ci}\left (x \right )+\cos \left (1\right )-\sin \left (1\right )\right ) \cos \left (x \right )+\left (4 \,\operatorname {Si}\left (1\right )-4 \,\operatorname {Si}\left (x \right )+\cos \left (1\right )+\sin \left (1\right )\right ) \sin \left (x \right )+4 \ln \left (x \right )-1 \]

✓ Solution by Mathematica

Time used: 60.029 (sec). Leaf size: 63

DSolve[{x*D[y[x],{x,3}]+x*D[y[x],x]==4,{y[1]==0,Derivative[1][y][1]==1,Derivative[2][y][1]==-1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\left (\cos (1-K[2])+\sin (1-K[2])+4 \operatorname {CosIntegral}(K[2]) \sin (K[2])-4 \operatorname {CosIntegral}(1) \sin (K[2])+\cos (K[2]) \int _1^{K[2]}-\frac {4 \sin (K[1])}{K[1]}dK[1]\right )dK[2] \]