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Problem 2: Separation of Variables Find the IVP solution for the following differential equation by separation of variables: DE: 2t dy = y^2(1 + 2t^2) dt IC: y(1) = -1 Solution: a. Begin by separating the variables: _ dy = dt b. Integrate both sides. Don't forget the constant of integration! Tips: ? x^n dx = (x^(n+1))/(n+1) + c (provided n ? 1): ? 1/x dx = ln|x| + c c. Before proceeding to an explicit solution, find c, the constant of integration, now using the IC: y(1) = -1 Thus, the specific solution matching the initial condition is: d. Flip both sides to find an explicit solution for y. y = ____

          Problem 2: Separation of Variables
Find the IVP solution for the following differential equation by separation of variables:
DE: 2t dy = y^2(1 + 2t^2) dt
IC: y(1) = -1
Solution: a. Begin by separating the variables:
_____ dy = __________ dt
b. Integrate both sides. Don't forget the constant of integration!
Tips: ? x^n dx = (x^(n+1))/(n+1) + c (provided n ? 1):
? 1/x dx = ln|x| + c
c. Before proceeding to an explicit solution, find c, the constant of integration, now using the IC: y(1) = -1
Thus, the specific solution matching the initial condition is:
d. Flip both sides to find an explicit solution for y.
y = __________

Problem 2: Separation of Variables
Find the IVP solution for the following differential equation by separation of variables:
DE: 2t dy = y^2(1 + 2t^2) dt
IC: y(1) = -1
Solution: a. Begin by separating the variables:
dy = dt
b. Integrate both sides. Don't forget the constant of integration!
Tips: ? x^n dx = (x^(n+1))/(n+1) + c (provided n ? 1):
? 1/x dx = ln|x| + c
c. Before proceeding to an explicit solution, find c, the constant of integration, now using the IC: y(1) = -1
Thus, the specific solution matching the initial condition is:
d. Flip both sides to find an explicit solution for y.
y =

Added by Brent F.

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