69.1.77 problem 122

Internal problem ID [14230]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 122
Date solved : Tuesday, January 28, 2025 at 06:22:44 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x y^{\prime \prime }-y^{\prime }&=x^{2} {\mathrm e}^{x} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 17

dsolve([x*diff(y(x),x$2)-diff(y(x),x)=x^2*exp(x),y(0) = -1, D(y)(0) = 0],y(x), singsol=all)
 

\[ y = \left (x -1\right ) {\mathrm e}^{x}+\frac {c_{1} x^{2}}{2} \]

✓ Solution by Mathematica

Time used: 0.404 (sec). Leaf size: 54

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

\begin{align*} y(x)\to e^x (x-1) \\ y(x)\to \int _1^x\left (c_1+e^{K[1]}\right ) K[1]dK[1]-\int _1^0\left (c_1+e^{K[1]}\right ) K[1]dK[1]-1 \\ \end{align*}