TU 02 :
Mathema-cal
and Analysis Tools for Physics – 1
Tutorials
Claire DARRAUD (UNILIM - France)
Copyright no,ce : This material can be freely used within the E.O.L.E.S. TEMPUS PROJECT consor,um. Explicit authorisa,on of the authors is required for its use outside this consor,um EOLES TEMPUS PROJECT. This
project has been funded with support from the European Commission. This publica,on reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the
informa,on contained therein.
Differen,al Equa,ons solving
(2nd order, linear, with constant coefficients
and second member)
BACK TO THE COURSE
IV - EXAMPLES. FIND ALL THE SOLUTIONS TO:
1/
2/
3/
d2 y
dy
+ ( 2m + 1)
+ 2my = e
4/ Let (Eq.) be with a parameter m (m ∈ │R ):
2
dx
dx
a) Find the solu,on to (Eq.) when m = 0.
Ini,al Condi,ons (IC) : y(0) = 0 = y'(0).
b) Find m that corresponds to €
Δ = 0.
Then, find the solu,on to (Eq.) with the same IC.
−2 x
TU2 Differen,al Equa,ons solving – claire.darraud@unilim.fr
Copyright no,ce : This material can be freely used within the E.O.L.E.S. TEMPUS PROJECT consor,um. Explicit authorisa,on of the authors is required for its use outside this consor,um EOLES TEMPUS PROJECT. This
project has been funded with support from the European Commission. This publica,on reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the
informa,on contained therein.
2
Differen,al Equa,ons solving
(2nd order, linear, with constant coefficients
and second member)
BACK TO THE COURSE
IV - EXAMPLES. FIND ALL THE SOLUTIONS TO:
1/
à y(t) = C1 et + C2 e-2t
2/
à y(t) = e-t (A cos2t+ B sin2t) = C e-t cos(2t – φ)
3/
à y(t) = e-2t(C1 + C2t)
TU2 Differen,al Equa,ons solving – claire.darraud@unilim.fr
Copyright no,ce : This material can be freely used within the E.O.L.E.S. TEMPUS PROJECT consor,um. Explicit authorisa,on of the authors is required for its use outside this consor,um EOLES TEMPUS PROJECT. This
project has been funded with support from the European Commission. This publica,on reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the
informa,on contained therein.
3
Differen,al Equa,ons solving
(2nd order, linear, with constant coefficients
and second member)
BACK TO THE COURSE
IV - EXAMPLES. FIND ALL THE SOLUTIONS TO:
d2 y
4/ Let (Eq.) be with a parameter m (m ∈ │R ):
dx2
a) Find the solu,on to (Eq.) when m = 0.
Ini,al Condi,ons (IC) : y(0) = 0 = y'(0).
à y(x) = ½
(e-2x +
1) -
e-x
(
dy
) dx + 2my = e
+ 2m + 1
−2 x
€
b) Find m that corresponds to Δ = 0.
Then, find the solu,on to (Eq.) with the same IC.
à m = ½ and y(x) = (x – 1) e-x + e-2x
TU2 Differen,al Equa,ons solving – claire.darraud@unilim.fr
Copyright no,ce : This material can be freely used within the E.O.L.E.S. TEMPUS PROJECT consor,um. Explicit authorisa,on of the authors is required for its use outside this consor,um EOLES TEMPUS PROJECT. This
project has been funded with support from the European Commission. This publica,on reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the
informa,on contained therein.
4
Differen,al Equa,ons solving
(2nd order, linear, with constant coefficients
and second member)
NEED SOME HELP ?
TU2 Differen,al Equa,ons solving – claire.darraud@unilim.fr
Copyright no,ce : This material can be freely used within the E.O.L.E.S. TEMPUS PROJECT consor,um. Explicit authorisa,on of the authors is required for its use outside this consor,um EOLES TEMPUS PROJECT. This
project has been funded with support from the European Commission. This publica,on reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the
informa,on contained therein.
5
Differen,al Equa,ons solving
(2nd order, linear, with constant coefficients
and second member)
NEED SOME HELP ?
TU2 Differen,al Equa,ons solving – claire.darraud@unilim.fr
Copyright no,ce : This material can be freely used within the E.O.L.E.S. TEMPUS PROJECT consor,um. Explicit authorisa,on of the authors is required for its use outside this consor,um EOLES TEMPUS PROJECT. This
project has been funded with support from the European Commission. This publica,on reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the
informa,on contained therein.
6
Differen,al Equa,ons solving
(2nd order, linear, with constant coefficients
and second member)
NEED SOME HELP ?
TU2 Differen,al Equa,ons solving – claire.darraud@unilim.fr
Copyright no,ce : This material can be freely used within the E.O.L.E.S. TEMPUS PROJECT consor,um. Explicit authorisa,on of the authors is required for its use outside this consor,um EOLES TEMPUS PROJECT. This
project has been funded with support from the European Commission. This publica,on reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the
informa,on contained therein.
7
Differen,al Equa,ons solving
(2nd order, linear, with constant coefficients
and second member)
NEED SOME HELP ?
TU2 Differen,al Equa,ons solving – claire.darraud@unilim.fr
Copyright no,ce : This material can be freely used within the E.O.L.E.S. TEMPUS PROJECT consor,um. Explicit authorisa,on of the authors is required for its use outside this consor,um EOLES TEMPUS PROJECT. This
project has been funded with support from the European Commission. This publica,on reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the
informa,on contained therein.
8
Differen,al Equa,ons solving
(2nd order, linear, with constant coefficients
and second member)
NEED SOME HELP ?
C1 and C2 could be found with IC
TU2 Differen,al Equa,ons solving – claire.darraud@unilim.fr
Copyright no,ce : This material can be freely used within the E.O.L.E.S. TEMPUS PROJECT consor,um. Explicit authorisa,on of the authors is required for its use outside this consor,um EOLES TEMPUS PROJECT. This
project has been funded with support from the European Commission. This publica,on reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the
informa,on contained therein.
9