grandes-ecoles 2013 QIV.B

grandes-ecoles · France · centrale-maths1__psi Second order differential equations Properties of special function solutions

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
Using question II.D, show by induction that for all integer $n \geqslant 1$ the function $\varphi _ { n }$ is strictly positive on $] 0 , \alpha _ { 0 } [$.

We introduce the differential equation

$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$

Using question II.D, show by induction that for all integer $n \geqslant 1$ the function $\varphi _ { n }$ is strictly positive on $] 0 , \alpha _ { 0 } [$.

Paper Questions

QI.A.1 QI.A.2 QI.B QI.C QI.D QII.A QII.B QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.D QII.E.1 QII.E.2 QII.E.3 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2