dx/dt=4(x^2+1), x(Pi/4)=1 Differential Equations Help!?
Ive tried working the problem out and i got to tan(x)=4T+C. They i applied the initial value and had Tan(1) = 4(pi/4)+C
I couldnt come up with an exact answer so that lead me to beleive it was wrong. Any help. Was i even on the right path to solving this problem or was i completely off? Thanks
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dx/dt = 4 * (x^2 + 1)
dx / (1 + x^2) = 4dt
arctan(x) = 4t + C
x = tan(4t + C)
x(t) = tan(4t + C)
t = pi/4
x(pi/4) = 1
x(pi/4) = tan(4 * (pi/4) + C)
1 = tan(pi + C)
tan(pi/4) = tan(pi + C)
pi/4 = pi + C
-3pi/4 = C
x(t) = tan(4t - 3pi/4)
Find the general solution by separating the variables then integrating:
dx / dt = 4(x² + 1)
dx / (x² + 1) = 4 dt
∫ 1 / (x² + 1) dx = 4 ∫ 1 dt
tanˉ¹x = 4t + C
x = tan(4t + C)
Find the particular solution by solving for the constant:
When t = π / 4, x = 1
tan(π + C) = 1
π + C = tanˉ¹1
π + C = π / 4
C = -3π / 4
x = tan(4t - 3π / 4)
x = tan(4t + π / 4)