15.8.45 problem 48

Internal problem ID [3048]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 48
Date solved : Monday, January 27, 2025 at 07:11:47 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

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

With initial conditions

\begin{align*} x \left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 16

dsolve([diff(x(theta),theta)=x(theta)+x(theta)^2*exp(theta),x(0) = 2],x(theta), singsol=all)
 

\[ x \left (\theta \right ) = -\frac {2 \,{\mathrm e}^{\theta }}{{\mathrm e}^{2 \theta }-2} \]

✓ Solution by Mathematica

Time used: 0.233 (sec). Leaf size: 19

DSolve[{D[ x[\[Theta]], \[Theta] ]==x[\[Theta]]+x[\[Theta]]^2*Exp[\[Theta]],{x[0]==2}},x[\[Theta]],\[Theta],IncludeSingularSolutions -> True]
 

\[ x(\theta )\to -\frac {2 e^{\theta }}{e^{2 \theta }-2} \]