75.21.4 problem 699

Internal problem ID [17173]
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 : 699
Date solved : Tuesday, January 28, 2025 at 09:56:07 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+{x^{\prime }}^{2}+x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 61

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

\begin{align*} -2 \left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_{1} -4 \textit {\_a}}}d \textit {\_a} \right )-t -c_{2} &= 0 \\ 2 \left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_{1} -4 \textit {\_a}}}d \textit {\_a} \right )-t -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.411 (sec). Leaf size: 308

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

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-e^{2 K[1]} K[1]dK[1]}}dK[2]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-e^{2 K[1]} K[1]dK[1]}}dK[3]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{K[2]}}{\sqrt {2 \int _1^{K[2]}-e^{2 K[1]} K[1]dK[1]-c_1}}dK[2]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-e^{2 K[1]} K[1]dK[1]}}dK[2]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[3]}}{\sqrt {2 \int _1^{K[3]}-e^{2 K[1]} K[1]dK[1]-c_1}}dK[3]\&\right ][t+c_2] \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-e^{2 K[1]} K[1]dK[1]}}dK[3]\&\right ][t+c_2] \\ \end{align*}