Solve the initial value problem dy/dx=8(x^3)*e^(-2y) y(1)=0

Elsa Brewer 2022-07-28 Answered
Solve the initial value problem
d y / d x = 8 ( x 3 ) e 2 y
y(1)=0
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Answers (2)

Kaiden Weeks
Answered 2022-07-29 Author has 14 answers
d y d x = 8 x 3 e 2 y
e 2 y d y = 8 x 3 d x
e 2 y d y = 8 x 3 d x
1 2 e 2 y = 2 x 4 + C
y(1)=0
1 2 e 2 0 = 2 ( 1 ) 4 + C
1 2 = 2 + C , C = 3 2
1 2 e 2 y = 2 x 4 3 2
e 2 y = 4 x 4 3
2 y = ln ( 4 x 4 3 )
y = 1 2 ln ( 4 x 4 3 )
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Ethen Blackwell
Answered 2022-07-30 Author has 3 answers
e 2 y d y = 8 x 3 d x
1 2 e 2 y = 2 x 4 + c
y = 1 2 ln ( 4 x 4 + A )
for the initial value y(1)=0 ie. y = 0, x = 1,
0 = 0.5 ln ( 4 + A )
ln ( 4 + A ) = 0
e 0 = 1 = 4 + A
A = 3
thus the solution is:
y = 1 2 ln ( 4 x 3 4 )
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