equações diferenciais

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❡ ❚❡❝♥♦❧♦❣✐❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ❡♠ ❆❧❣✉♥s
❈♦♥t❡①t♦s ❍✐stór✐❝♦s ❡ ❘❡❛✐s

❆✉t♦r✿

❊❧✐❛s ❈❛♠♣♦s ❞❛ ❙✐❧✈❛

❖r✐❡♥t❛❞♦r✿

❙❡❧♠❛ ❍✳ ❞❡ ❏❡s✉s ◆✐❝♦❧❛

❉✐s❝✐♣❧✐♥❛✿ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞♦ ❈✉rs♦ ❆

❈✉rs♦✿

▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛

Pr♦❢❡ss♦r❡s ❘❡s♣♦♥sá✈❡✐s✿

❑❛r✐♥❛ ❙❝❤✐❛❜❡❧ ❙✐❧✈❛
❚♦♠❛s ❊❞s♦♥ ❇❛rr♦s
❱❡r❛ ▲ú❝✐❛ ❈❛r❜♦♥❡

❙ã♦ ❈❛r❧♦s✱ ✶✸ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✶✳

❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ❡♠ ❆❧❣✉♥s
❈♦♥t❡①t♦s ❍✐stór✐❝♦s ❡ ❘❡❛✐s

❆✉t♦r✿

❊❧✐❛s ❈❛♠♣♦s ❞❛ ❙✐❧✈❛

❖r✐❡♥t❛❞♦r✿

❙❡❧♠❛ ❍✳ ❞❡ ❏❡s✉s ◆✐❝♦❧❛

❉✐s❝✐♣❧✐♥❛✿ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞♦ ❈✉rs♦ ❆

❈✉rs♦✿

▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛

Pr♦❢❡ss♦r❡s ❘❡s♣♦♥sá✈❡✐s✿

❑❛r✐♥❛ ❙❝❤✐❛❜❡❧ ❙✐❧✈❛
❚♦♠❛s ❊❞s♦♥ ❇❛rr♦s
❱❡r❛ ▲ú❝✐❛ ❈❛r❜♦♥❡

■♥st✐t✉✐çã♦✿

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❡ ❚❡❝♥♦❧♦❣✐❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❙ã♦ ❈❛r❧♦s✱ ✶✸ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✶✳

❊❧✐❛s ❈❛♠♣♦s ❞❛ ❙✐❧✈❛

❙❡❧♠❛ ❍✳ ❞❡ ❏❡s✉s ◆✐❝♦❧❛

❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣❡❧❛ ❢♦rç❛ ❡ s❛ú❞❡ q✉❡ ❡❧❡ t❡♠ ❞❛❞♦ ❛ ♠✐♠ ♣❛r❛
❝♦♥❝❧✉✐r ❡st❡ ❚❈❈✳ ❆❣r❛❞❡ç♦ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ Pr♦❢❡ss♦r❛ ❖r✐❡♥t❛❞♦r❛ ❙❡❧♠❛ ❞❡ ❏❡s✉s ♣❡❧❛ s✉❛ ❛t❡♥çã♦ ❡ ♣❛❝✐ê♥❝✐❛ ♥♦s ♥♦ss♦s ❡♥❝♦♥tr♦s s❡♠❛♥❛✐s✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠✱ ❛♦s Pr♦❢❡ss♦r❡s
❞♦ ❉▼✱ q✉❡ t❛♥t♦ ♠❡ ❛❥✉❞❛r❛♠ ❛♦ ❧♦♥❣♦ ❞❡ss❡s ❛♥♦s ♥❛ ❝♦♥str✉çã♦ ❞♦ ♠❡✉ ❝♦♥❤❡❝✐♠❡♥t♦
♠❛t❡♠át✐❝♦✳ ❆ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❡ t❛♠❜é♠✱ ♣♦r ♠✐♥❤❛ ❛✉sê♥❝✐❛ ❡♠ ♥♦ss❛s r❡✉♥✐õ❡s ❢❛♠✐❧✐❛r❡s✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♥♦✐✈❛ ●❛❜r✐❡❧❛ ♣♦r s✉♣♦rt❛r ❛ ♠✐♥❤❛ ❛✉sê♥❝✐❛

❞✉r❛♥t❡ ❛ s❡♠❛♥❛ ❡ ♠❡✉ ♠❛✉ ❤✉♠♦r✱ q✉❛♥❞♦ ❡✉ ❡♥❢r❡♥t❛✈❛ ♣r♦❜❧❡♠❛s ♥❛ ❝♦♥str✉çã♦
❞❡st❡ tr❛❜❛❧❤♦✳ ❊ ✜♥❛❧✐③❛♥❞♦ ❛❣r❛❞❡ç♦ ❛ ♠❡✉ ❛♠✐❣♦ ❡ ✐r♠ã♦ ❇r✉♥♦ ▼❡♥❞❡s ♣♦r t♦❞❛ s✉❛
❛❥✉❞❛✱ ❛♦ ♠❡ ❡♥s✐♥❛r ❛ ♠❛♥✐♣✉❧❛r ♦ ❚❊❳✳

❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❆ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡ s✉❛s ❛♣❧✐❝❛çõ❡s✱ ❡♥✈♦❧✈❡♥❞♦ ❢❛t♦s ❤✐stór✐❝♦s ❡ r❡❛✐s✳

❙❡rá ❞❛❞♦ ê♥❢❛s❡ às ❡q✉❛çõ❡s ❞✐❢❡✲

r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❡ ❞❡✐①❛r❡♠♦s ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ s❡❣✉♥❞❛
♦r❞❡♠ ♣❛r❛ ♦ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦