75.21.9 problem 704

Internal problem ID [17178]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 16. The method of isoclines for differential equations of the second order. Exercises page 158
Problem number : 704
Date solved : Tuesday, January 28, 2025 at 09:56:14 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.309 (sec). Leaf size: 30

dsolve(diff(x(t),t$2)+(x(t)+2)*diff(x(t),t)=0,x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\left (-c_{1} \sqrt {2}+\tanh \left (\frac {\left (c_{2} +t \right ) \sqrt {2}}{2 c_{1}}\right )\right ) \sqrt {2}}{c_{1}} \]

✓ Solution by Mathematica

Time used: 0.138 (sec). Leaf size: 108

DSolve[D[x[t],{t,2}]+(x[t]+2)*D[x[t],t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {1}{2} K[1]^2-2 K[1]+c_1}dK[1]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {1}{2} K[1]^2-2 K[1]-c_1}dK[1]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {1}{2} K[1]^2-2 K[1]+c_1}dK[1]\&\right ][t+c_2] \\ \end{align*}