Answer:
(a) (y^5)/5 + y^4 = (t^3)/3 + 7t + C
(b) y = arctan(t(lnt - 1) + C)
(c) y = -1/ln|0.09(t + 1)²/t|
Step-by-step explanation:
(a) dy/dt = (t^2 + 7)/(y^4 - 4y^3)
Separate the variables
(y^4 - 4y^3)dy = (t^2 + 7)dt
Integrate both sides
(y^5)/5 + y^4 = (t^3)/3 + 7t + C
(b) dy/dt = (cos²y)lnt
Separate the variables
dy/cos²y = lnt dt
Integrate both sides
tany = t(lnt - 1) + C
y = arctan(t(lnt - 1) + C)
(c) (t² + t) dy/dt + y² = ty², y(1) = -1
(t² + t) dy/dt = ty² - y²
(t² + t) dy/dt = y²(t - 1)
(t² + t)/(t - 1)dy/dt = y²
Separating the variables
(t - 1)dt/(t² + t) = dy/y²
tdt/(t² + t) - dt/(t² + t) = dy/y²
dt/(t + 1) - dt/(t(t + 1)) = dy/y²
dt/(t + 1) - dt/t + dt/(t + 1) = dy/y²
Integrate both sides
ln(t + 1) - lnt + ln(t + 1) + lnC = -1/y
2ln(t + 1) - lnt + lnC = -1/y
ln|C(t + 1)²/t| = -1/y
y = -1/ln|C(t + 1)²/t|
Apply y(1) = -1
-1 = ln|C(1 + 1)²/1|
-1 = ln(4C)
4C = e^(-1)
C = (1/4)e^(-1) ≈ 0.09
y = -1/ln|0.09(t + 1)²/t|
