Evaluate the integrals using either by parts method or

Question

Question 1: Evaluate the following integrals using either by parts method or substitution

a) y = ∫6x² e^2X+10 dx

b) y = ∫ -3x^{2 5}√(10 - x³)dx

Question 2: Solve the following ODE using an integrating factor:

xy^' - 7y = 4x¹⁰

Question 3: Show, if:

y(x) = -e^-x - 42x - 42

is a solution to the ODE:

y' -1^y_Y = 42x

Question 4:

Show, if:

y(t) = 1/10.e^10x + sin²(x) + cos²(x)

is a solution to the ODE:

101/10. y' -(y - y'') + 1 =0

Hint: Can the equation be simplified?

Question 5:

a) Plot the following vector field from -5 ≤ x ≤ 5, -5 ≤ y ≤ 5:

y' = sin(x) + cos(x)

b) Note this vector field contains all possible solutions. Now find out the particular solution passing through the point P(1,1) and plot this curve on the vector field graph.

Please use a different colour for the particular solution. Do not try to calculate the particular solution, but find it by carefully investigating the vector field!

Question 6: Find the general solution to the following second order linear constant coefficient homogenous ODEs

6y" + 4y' - 10y = 0

Question 7: 'Practical' application. For both parts just find the general solution to the ODE

a) Mechanical: Given the diagram below construct an ODE and solve for the displacement y mass:05 kg  Spring constant:3.125 N/m, Damping Constant:2.5 Ns/m

b) Electrical: Given the RLC network below construct an ODE and solve for the current i(t)

Hint: CV = ∫idt,       V = Ldi/dt,   V = iR

Question 8: Find the ODE that satisfies the solution:

y(x) = 5 cos(4x) + 5sin(4x)

In the form

Ay" + By' + Cy = 0

Hint: Just pick a value for one of the coefficients when you get stuck.

Question 9: Find the general solution the following ODE

5y" - y' - 6y = sin(3x) - 3e^-x

Clearly show working for both yk and yp.

Question 10:

Mixing Tanks T₁ and T₂ contain 400 Litres of water each. T₁ is pure water, however in T₂ there is 60 Kg of secret sauce dissolved uniformly. T₁ has 20 Litres/min, of pure water coming from the outside and has 16 Litres/min coming from T₂. T₂ has 36 Litres/min coming from T₁ and has 20 Litres/min exiting to the outside.

Assume the tanks have uniform mixing.

How much secret sauce is left in T₂ after one hour?