2.1.14.5 ✓ Maple. Time used: 0.001 (sec). Leaf size: 18
ode:=1+2*x(t)+(-t^2+4)*diff(x(t),t) = 0;
dsolve(ode,x(t), singsol=all);
\begin{align*} x = -\frac {1}{2}+\frac {\sqrt {t -2}\, c_1}{\sqrt {t +2}} \end{align*}
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 1+2 x \left (t \right )+\left (-t^{2}+4\right ) \left (\frac {d}{d t}x \left (t \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d t}x \left (t \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x \left (t \right )=\frac {-1-2 x \left (t \right )}{-t^{2}+4} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d t}x \left (t \right )}{-1-2 x \left (t \right )}=\frac {1}{-t^{2}+4} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {\frac {d}{d t}x \left (t \right )}{-1-2 x \left (t \right )}d t =\int \frac {1}{-t^{2}+4}d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (-1-2 x \left (t \right )\right )}{2}=\frac {\ln \left (t +2\right )}{4}-\frac {\ln \left (t -2\right )}{4}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \left (t \right ) \\ {} & {} & \left \{x \left (t \right )=\frac {-{\mathrm e}^{4 \mathit {C1}} t +\sqrt {{\mathrm e}^{4 \mathit {C1}} t^{2}-4 \,{\mathrm e}^{4 \mathit {C1}}}-2 \,{\mathrm e}^{4 \mathit {C1}}}{2 \,{\mathrm e}^{4 \mathit {C1}} \left (t +2\right )}, x \left (t \right )=-\frac {{\mathrm e}^{4 \mathit {C1}} t +\sqrt {{\mathrm e}^{4 \mathit {C1}} t^{2}-4 \,{\mathrm e}^{4 \mathit {C1}}}+2 \,{\mathrm e}^{4 \mathit {C1}}}{2 \,{\mathrm e}^{4 \mathit {C1}} \left (t +2\right )}\right \} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & \left \{x \left (t \right )=\frac {-t -2+{\mathrm e}^{-4 \mathit {C1}} \sqrt {{\mathrm e}^{4 \mathit {C1}} \left (t^{2}-4\right )}}{2 t +4}, x \left (t \right )=\frac {-{\mathrm e}^{-4 \mathit {C1}} \sqrt {{\mathrm e}^{4 \mathit {C1}} \left (t^{2}-4\right )}-t -2}{2 t +4}\right \} \end {array} \]