83.27.6 problem 6

Internal problem ID [19352]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (A) at page 104
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 01:35:12 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 30

dsolve(diff(y(x),x$2)+2*exp(x)*diff(y(x),x)+2*exp(x)*y(x)=x^2,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (3 c_{2} +\int \left (x^{3}+3 c_{1} \right ) {\mathrm e}^{2 \,{\mathrm e}^{x}}d x \right ) {\mathrm e}^{-2 \,{\mathrm e}^{x}}}{3} \]

✓ Solution by Mathematica

Time used: 60.021 (sec). Leaf size: 44

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

\[ y(x)\to e^{-2 e^x} \left (\int _1^x\frac {1}{3} e^{2 e^{K[1]}} \left (K[1]^3+3 c_1\right )dK[1]+c_2\right ) \]