Find derivatives with respect to parameter or initial condition Find stated derivatives with respect to parameter or initial condition:
a) $x'=x+\mu(t+x^2), x(0)=1 ;$ find $\frac{\partial x}{\partial \mu}|_{\mu=0}$ ,
b) $x'=2t+\mu x^2, x(0)=\mu -1;$ find $\frac{\partial x}{\partial \mu}|_{\mu=0}$ ,
c) $x'=x+x^2 +tx^3, x(2)=x_0 ;$ find $\frac{\partial x}{\partial \mu}|_{\mu=0}$ ,
Please help me with at least one of them, and I will try to make the others. I really don't know what to do in here. 
 A: Regarding the first exercise.
Given a differential equation 
$$
\dot{x}=f(x,\mu)
$$
the sensitivity of $x(t,\mu)$ regarding $\mu$ or 
$$
\frac{\partial x}{\partial\mu}
$$
can be calculated as
$$
\frac{d}{dt}\left(\frac{\partial x}{\partial\mu}\right)=\frac{\partial f}{\partial x}\frac{\partial x}{\partial\mu}+\frac{\partial f}{\partial\mu}
$$
so calling $\delta_{\mu}=\frac{\partial x}{\partial\mu}$ we have
the sensitivity equation
$$
\dot{\delta}_{\mu}=\frac{\partial f}{\partial x}\delta_{\mu}+\frac{\partial f}{\partial\mu}
$$
In the case of 
$$
\dot{x}=x+\mu(t+x^{2}),\;x(0)=1
$$
we have
$$
\begin{array}{rcl}
\frac{\partial f}{\partial x} & = & 1+2\mu x\\
\frac{\partial f}{\partial\mu} & = & t+x^{2}
\end{array}
$$
and the final system to solve is
$$
\begin{array}{rcl}
\dot{\delta_{\mu}} & = & (1+2\mu x)\delta_{\mu}+t+x^{2}, \; \delta_{\mu}(0) = 0\\
\dot{x} & = & x+\mu(t+x^{2}),\;x(0)=1
\end{array}
$$
focusing the case of $\mu=0$ we have to solve 
$$
\begin{array}{rcl}
\dot{\delta}_{0} & = & \delta_{0}+t+x^{2}, \; \delta_0 = 0\\
\dot{x} & = & x,\;x(0)=1
\end{array}
$$
NOTE
If 
$$
x(0) = g(\mu)\Rightarrow \delta_{\mu}(0) = \frac{\partial g}{\partial \mu}
$$
