2.1.1.4 ✓ Maple. Time used: 0.010 (sec). Leaf size: 14

ode:=diff(x(t),t) = 3*t^2+4*t; 
ic:=[x(1) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\begin{align*} x = t^{3}+2 t^{2}-3 \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}x \left (t \right )=3 t^{2}+4 t , x \left (1\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d t}x \left (t \right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (\frac {d}{d t}x \left (t \right )\right )d t =\int \left (3 t^{2}+4 t \right )d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & x \left (t \right )=t^{3}+2 t^{2}+\mathit {C1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} x \left (1\right )=0 \\ {} & {} & 0=\mathit {C1} +3 \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \textit {\_C1} \\ {} & {} & \mathit {C1} =-3 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \textit {\_C1} =-3\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & x \left (t \right )=t^{3}+2 t^{2}-3 \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & x \left (t \right )=t^{3}+2 t^{2}-3 \end {array} \]