56.2.20 problem 19

Internal problem ID [8824]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 19
Date solved : Monday, January 27, 2025 at 05:03:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y^{\prime }-y x -x^{3}+1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 67

dsolve(diff(y(x),x$2)-diff(y(x),x)-x*y(x)-x^3+1=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{\frac {x}{2}} \left (-\operatorname {AiryAi}\left (x +\frac {1}{4}\right ) \pi \left (\int \left (x^{3}-1\right ) \operatorname {AiryBi}\left (x +\frac {1}{4}\right ) {\mathrm e}^{-\frac {x}{2}}d x \right )+\operatorname {AiryBi}\left (x +\frac {1}{4}\right ) \pi \left (\int \left (x^{3}-1\right ) \operatorname {AiryAi}\left (x +\frac {1}{4}\right ) {\mathrm e}^{-\frac {x}{2}}d x \right )+c_{2} \operatorname {AiryAi}\left (x +\frac {1}{4}\right )+c_{1} \operatorname {AiryBi}\left (x +\frac {1}{4}\right )\right ) \]

✓ Solution by Mathematica

Time used: 3.812 (sec). Leaf size: 107

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

\[ y(x)\to e^{x/2} \left (\operatorname {AiryAi}\left (x+\frac {1}{4}\right ) \int _1^x-e^{-\frac {K[1]}{2}} \pi \operatorname {AiryBi}\left (K[1]+\frac {1}{4}\right ) \left (K[1]^3-1\right )dK[1]+\operatorname {AiryBi}\left (x+\frac {1}{4}\right ) \int _1^xe^{-\frac {K[2]}{2}} \pi \operatorname {AiryAi}\left (K[2]+\frac {1}{4}\right ) \left (K[2]^3-1\right )dK[2]+c_1 \operatorname {AiryAi}\left (x+\frac {1}{4}\right )+c_2 \operatorname {AiryBi}\left (x+\frac {1}{4}\right )\right ) \]