69.1.40 problem 59

Internal problem ID [14193]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 59
Date solved : Tuesday, January 28, 2025 at 06:21:26 AM
CAS classification : [_linear]

\begin{align*} \left (-x^{2}+x \right ) y^{\prime }+\left (2 x^{2}-1\right ) y-a \,x^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 39

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

\[ y = -\left (a \,{\mathrm e}^{2 x -2} \left (x -1\right ) \operatorname {Ei}_{1}\left (2 x -2\right )-c_{1} \left (x -1\right ) {\mathrm e}^{2 x}-a \right ) x \]

✓ Solution by Mathematica

Time used: 0.269 (sec). Leaf size: 91

DSolve[(x-x^2)*D[y[x],x]+(2*x^2-1)*y[x]-a*x^3==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {1-2 K[1]^2}{K[1]-K[1]^2}dK[1]\right ) \left (\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1-2 K[1]^2}{K[1]-K[1]^2}dK[1]\right ) K[2]^2}{1-K[2]}dK[2]+c_1\right ) \]