Question

Resolva o problema de valor inicial a seguir

Answer 1

Resposta:

(e^(2y) -y)*cos(x) *dy/dx =e^(y) * sen(2x)

[e^(2y)-y]/e^(y)   dy  = sen(2x)/cos(x)  dx

e^(y)- y*e^(-y)   dy  = 2*sen(x)*cos(x)/cos(x)  dx

e^(y)- y*e^(-y)   dy  = 2*sen(x)  dx

∫  e^(y)- y*e^(-y)   dy  = 2* ∫ sen(x)  dx

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∫ y*e^(-y)   dy

Fazendo por partes

u=y  ==>du=dy

dv =e^(-y)   dy   ==>∫ dv =∫e^(-y)   dy  ==>v=-e^(-y)

∫ y*e^(-y)   dy  =-y * e^(-y) - ∫-e^(-y) dy

∫ y*e^(-y)   dy  =-y * e^(-y) +∫e^(-y) dy

∫ y*e^(-y)   dy  =-y * e^(-y) -e^(-y)  =-e^(-y) *(y +1) + c

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∫  e^(y) dy - ∫  y*e^(-y)   dy  = 2* ∫ sen(x)  dx

=> e^(y) +e^(-y) *(y +1) =-2*cos(x)  + c

para y(0)=0

=> e^(0) +e^(-0) *(0 +1) =-2*cos(0)  + c

1+1=-2*1 + c  ==>c=4

e^(y) +e^(-y) *(y +1) =-2*cos(x)  + 4