Have a tough time understanding calculus , in need of help please, appreciate you in advance <3
(i) Find the general solution of the Differential equation
𝑑2𝑦/𝑑𝑡2 = −4𝑡2 + 5𝑡 + 3
(ii)Hence, find the solution when 𝑡 = 0 , 𝑦 = 1 and 𝒅𝒚/𝒅𝒕= 1.
Hi @Maylin!
So, I'm not exactly sure what the answer is either but maybe this will help you. (Sorry if you still don't understand after this)
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follow these steps to determine the general solution y(t) using an integrating factor:
Calculate the integrating factor I(t). I ( t ) .
Multiply the standard form equation by I(t). I ( t ) .
Simplify the left-hand side to. ddt[I(t)y]. d d t [ I ( t ) y ] .
Integrate both sides of the equation.
Solve for y(t). y ( t ) .
(I really do apologize if you don't understand.)
y" = -4t^2 + 5t + 3
y' = -4/3 t^3 + 5/2 t^2 + 3t + c1
y = -1/3 t^4 + 5/6 t^3 + 3/2 t^2 + c1*t + c2
That is the general solution.
Using y'(0) = 1, we get c1 = 1
Using y(0) = 1, we get c2 = 1
so the particular solution is
t = -1/3 t^4 + 5/6 t^3 + 3/2 t^2 + t + 1