Consider the IVP y4 x2 y2 y1 2 Use Eulers iternative form

Consider the IVP y^4 = x^2 - y^2, y(1) = 2. Use Euler\'s iternative formula y_n + 1 = y_n + h f(x-n, y_n) to approximate y(3) with h = 1. The temperature of a roast placed in a 70 degree room is decreasing at a rate proportional to the difference between the temperature of the roast and room temperature. Initially, the roast is 200 degree. After 5 minutes, it is 160 degree. Find T(t), the temperature of the roast after t minutes Find the general solution of y^(4) - 2y\" + y = 0 Find the form of a particular solution for y\" - 9y = e^3x + x^2 using the method of undetermined coefficients. (You do not need to solve the differential equation-just find the form of y_p.)

Solution

Given problem is y\' = y² - x² and y(1) = 2

Also Euler Iterative formula is given by y_n+1 = y_n + h f (x_n,y_n).

Given x₀ =1 and y₀ = y(1) = 2 and f(x,) = y²-x²

So, y₁ = y₀ + h f(x₀,y₀) = 2 + 1 f(1,2) = 2 + (2² - 1²) = 2 +3 = 5

now, x₁ = x₀ + h = 1+1 =2

so y₂ = y₁ + h f(x₁,y₁) = 5 + 1 f(2,5) = 5 + (5² - 2²) = 5 +21 = 26

now, x₂ = x₁ + h = 2+1 =3

so y₃ = y₂ + h f(x₂,y₂) = 26 + 1 f(3,26) = 26 + (26² - 3²) = 26 +667 = 693

Thus y₃ = y(3) = 693.