83.41.24 problem 5 (xi)

Internal problem ID [19504]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise at end of chapter VIII. Page 141
Problem number : 5 (xi)
Date solved : Tuesday, January 28, 2025 at 08:32:47 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.191 (sec). Leaf size: 12

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

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

✓ Solution by Mathematica

Time used: 0.212 (sec). Leaf size: 16

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

\[ y(x)\to e^{-\frac {x^2}{2}}+1 \]