64.12.16 problem 16

Internal problem ID [13449]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 16
Date solved : Monday, March 31, 2025 at 07:58:12 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&={\mathrm e}^{x} \arcsin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 37

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+y(x) = exp(x)*arcsin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{x} \left (2 x^{2} \arcsin \left (x \right )+3 x \sqrt {-x^{2}+1}+4 c_1 x +\arcsin \left (x \right )+4 c_2 \right )}{4} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 45

ode=D[y[x],{x,2}]-2*D[y[x],x]+y[x]==Exp[x]*ArcSin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{4} e^x \left (2 x^2 \arcsin (x)+\arcsin (x)+3 \sqrt {1-x^2} x+4 c_2 x+4 c_1\right ) \]

✓ Sympy. Time used: 0.300 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - exp(x)*asin(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x \operatorname {asin}{\left (x \right )}}{2} + \frac {3 \sqrt {1 - x^{2}}}{4}\right ) + \frac {\operatorname {asin}{\left (x \right )}}{4}\right ) e^{x} \]