Solve y at t=1. Assume C1 = C2 = C3 = 1 (arbitrary constants)y''' + y' = t + 1. Use 2 decimal places.

To solve the given differential equation, we will use the method of undetermined coefficients.

Step 1: Find the complementary solution. The complementary solution is found by solving the homogeneous equation, which is obtained by setting the right-hand side of the differential equation to zero. In this case, the homogeneous equation is y''' + y' = 0.

The characteristic equation is obtained by substituting y = e^(rt) into the homogeneous equation: r^3 + r = 0.

Factoring out an r, we get: r(r^2 + 1) = 0.

This equation has three roots: r1 = 0, r2 = i, and r3 = -i.

Therefore, the complementary solution is: y_c(t) = C1 + C2cos(t) + C3sin(t), where C1, C2, and C3 are arbitrary constants.

Step 2: Find the particular solution. To find the particular solution, we assume that y_p(t) has the same form as the right-hand side of the differential equation, which is t + 1. Since the equation contains a polynomial of degree 1, we assume a particular solution of the form y_p(t) = At + B.

Substituting this into the differential equation, we get: (A + 1) + A = t + 1.

Equating the coefficients of like terms, we have: 2A + 1 = t.

Solving for A, we find A = 1/2.

Therefore, the particular solution is: y_p(t) = (1/2)t + B, where B is an arbitrary constant.

Step 3: Find the complete solution. The complete solution is the sum of the complementary and particular solutions: y(t) = y_c(t) + y_p(t).

Substituting the values of C1, C2, C3, and B as 1, the complete solution is: y(t) = 1 + cos(t) + sin(t) + (1/2)t.

Step 4: Find y at t = 1. To find y at t = 1, we substitute t = 1 into the complete solution: y(1) = 1 + cos(1) + sin(1) + (1/2)(1).

Using a calculator to evaluate the trigonometric functions and rounding to 2 decimal places, we find: y(1) ≈ 2.38.

Therefore, y at t = 1 is approximately 2.38.