grandes-ecoles 2016 QII.C.2

grandes-ecoles · France · centrale-maths1__pc Second order differential equations Solving homogeneous second-order linear ODE

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, and $F$ satisfies $F^{\prime} + AF = 0$ on $\mathbb{R}$. Deduce an expression for $F(x)$.
You may start by differentiating the function $x \mapsto -\frac{1}{8} \ln\left(1 + x^{2}\right) + \frac{\mathrm{i}}{4} \arctan x$.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, and $F$ satisfies $F^{\prime} + AF = 0$ on $\mathbb{R}$. Deduce an expression for $F(x)$.

You may start by differentiating the function $x \mapsto -\frac{1}{8} \ln\left(1 + x^{2}\right) + \frac{\mathrm{i}}{4} \arctan x$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QII.A QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIII.C.6 QIII.D.1 QIII.D.2 QIII.D.3 QIII.E.1 QIII.E.2 QIII.F