Internal problem ID [6403]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 19.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-y^{\prime }-x y-x^{3}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.091 (sec). Leaf size: 75
dsolve(diff(y(x),x$2)-diff(y(x),x)-x*y(x)-x^3+1=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\frac {x}{2}} \AiryAi \left (x +\frac {1}{4}\right ) c_{2}+{\mathrm e}^{\frac {x}{2}} \AiryBi \left (x +\frac {1}{4}\right ) c_{1}+\pi \,{\mathrm e}^{\frac {x}{2}} \left (\AiryBi \left (x +\frac {1}{4}\right ) \left (\int \left (x^{3}-1\right ) \AiryAi \left (x +\frac {1}{4}\right ) {\mathrm e}^{-\frac {x}{2}}d x \right )-\left (\int \left (x^{3}-1\right ) \AiryBi \left (x +\frac {1}{4}\right ) {\mathrm e}^{-\frac {x}{2}}d x \right ) \AiryAi \left (x +\frac {1}{4}\right )\right ) \]
✓ Solution by Mathematica
Time used: 2.215 (sec). Leaf size: 107
DSolve[y''[x]-y'[x]-x*y[x]-x^3+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{x/2} \left (\text {Ai}\left (x+\frac {1}{4}\right ) \int _1^x-e^{-\frac {K[1]}{2}} \pi \text {Bi}\left (K[1]+\frac {1}{4}\right ) \left (K[1]^3-1\right )dK[1]+\text {Bi}\left (x+\frac {1}{4}\right ) \int _1^xe^{-\frac {K[2]}{2}} \pi \text {Ai}\left (K[2]+\frac {1}{4}\right ) \left (K[2]^3-1\right )dK[2]+c_1 \text {Ai}\left (x+\frac {1}{4}\right )+c_2 \text {Bi}\left (x+\frac {1}{4}\right )\right ) \\ \end{align*}