83.8.22 problem 23

Internal problem ID [19148]
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. Misc examples on chapter II at page 25
Problem number : 23
Date solved : Tuesday, January 28, 2025 at 01:06:44 PM
CAS classification : [_separable]

\begin{align*} s^{\prime }+x^{2}&=x^{2} {\mathrm e}^{3 s} \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 19

dsolve(diff(s(x),x)+x^2=x^2*exp(3*s(x)),s(x), singsol=all)
 

\[ s \left (x \right ) = \frac {\ln \left (-\frac {1}{{\mathrm e}^{x^{3}} c_{1} -1}\right )}{3} \]

✓ Solution by Mathematica

Time used: 0.782 (sec). Leaf size: 132

DSolve[D[s[x],x]+x^2==x^2*Exp[3*s[x]],s[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} s(x)\to \log \left (-\sqrt [3]{-\frac {1}{2}} \sqrt [3]{1-\tanh \left (\frac {1}{2} \left (x^3+3 c_1\right )\right )}\right ) \\ s(x)\to \log \left (\frac {\sqrt [3]{1-\tanh \left (\frac {1}{2} \left (x^3+3 c_1\right )\right )}}{\sqrt [3]{2}}\right ) \\ s(x)\to \log \left (\frac {(-1)^{2/3} \sqrt [3]{1-\tanh \left (\frac {1}{2} \left (x^3+3 c_1\right )\right )}}{\sqrt [3]{2}}\right ) \\ s(x)\to 0 \\ s(x)\to -\frac {2 i \pi }{3} \\ s(x)\to \frac {2 i \pi }{3} \\ \end{align*}