3. Determine the function $f: R \rightarrow R$ that satisfies the conditions
a) $f$ is differentiable with a continuous derivative;
b) $f(0)=2013$;
c) $f^{\prime}(\mathrm{x})-\mathrm{f}(\mathrm{x})=e^{x} \cos x$, for any real number $x$.

Teodor Trișcă and Daniela Vicol, Botoșani, G.M Supplement
3. Deduce that $\left(e^{x} \cdot \sin x\right)^{\prime}=e^{x} \cdot(\sin x)^{\prime}+\left(e^{x}\right)^{\prime} \cdot \sin x=e^{x} \cdot \cos x+e^{x} \sin x \quad$ (1p).

Deduce also that $\left(e^{x} \cdot \sin x\right)^{\prime}-\left(e^{x} \cdot \sin x\right)=e^{x} \cos x \quad (2 \mathrm{p})$

Deduce that the primitive calculated must have the form $e^{x} \cdot \sin x+k \cdot e^{x}, x \in \mathbb{R} \quad (1 p)$

From the calibration condition, deduce that $k=2013 \quad (2 \mathrm{p})$

Obtain the primitive as $f(x)=e^{x} \sin x+2013 e^{x}, x \in \mathbb{R} \quad (1p)$