83.5.18 problem 18

Internal problem ID [19113]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (D) at page 16
Problem number : 18
Date solved : Tuesday, January 28, 2025 at 12:58:01 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }+x&=x \,{\mathrm e}^{\left (n -1\right ) y} \end{align*}

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 30

dsolve(diff(y(x),x)+x=x*exp( (n-1)*y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.397 (sec). Leaf size: 47

DSolve[D[y[x],x]+x==x*Exp[ (n-1)*y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\log \left ((n-1) \left (\frac {1}{n-1}-e^{\frac {1}{2} (n-1) \left (x^2+2 c_1\right )}\right )\right )}{n-1} \\ y(x)\to 0 \\ \end{align*}