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Eduardwww [97]
3 years ago
5

Find the solution of the differential equation that satisfies the given initial condition. (du)/(dt) = (2t + sec^2 t)/(2u), u(0)

= -1
Mathematics
1 answer:
Delicious77 [7]3 years ago
3 0
Ah okay so in differential equations you usually want the top variable isolated. To do this, multiply by dt and 2u and you get
2udu = 2t + { \sec(t) }^{2} dt
Now just integrate both sides. The integral of 2u with respect to u is u². The integral of (2t + sec²(t) with respect to t is t² + ∫sec²(t)dt. The last part is just tan(x) because d/dt(tan(t)) is sec²(t) so just integrating gets us back. Now we have
{u}^{2}   + c =  {t}^{2}  +  \tan(t)  + k
Where c and k are arbitrary constants. Subtracting c from k and you get
{u}^{2}  =  {t}^{2}  +  \tan(t)  + b
Where b is another constant. To find b, just plug in u(0) = -1 where u is -1 and t is 0. This becomes
1 =  \tan(0)  + b
tan(0) is 0 so b = 1. Take the plus or minus square root on both sides and you finally get
u =  \: { {t}^{2} }  +  \tan(t)  + 1
But Brainly didn't let me do but juat remember there is a plus or minus square root on the left.
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