83.40.5 problem 5

Internal problem ID [19477]
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 VIII (E) at page 140
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 01:45:57 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\frac {2}{1+{\mathrm e}^{x}} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 38

dsolve(diff(y(x),x$2)-y(x)=2/(1+exp(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 47

DSolve[D[y[x],{x,2}]-y[x]==2/(1+Exp[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 2 e^x \text {arctanh}\left (2 e^x+1\right )-e^{-x} \log \left (e^x+1\right )+c_1 e^x+c_2 e^{-x}-1 \]