7. (15pts) Consider the two-point boundary value problem u" + 2xu' x²u = x², u(0) = 1, u(1) = 0. - (a) Let h=1/4. Write out the difference equations explicitely, using centered-difference quotients for all derivatives. (b) Convert the equations to a linear system Ax = b, and then solve the system (either by hand or using MATLAB).

Write the given second order equation as its equivalent system of first order equations. u" + 4.5u' – 7.5u = -7 sin(3t), u(1) = -7.5, u'(1) = -6 Use v to represent the "velocity function", i.e. v = u'(t). Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.) u' = %3D Now write the system using matrices: u d + dt V V and the initial value for the vector valued function is: u(1) v(1)

The first-order DE is a linear equation if it has the following form: dy a(1)+a,(»)x= g(x)“ А dx B dx a(x)+a,(x)y = g) dy В dy •.. © a, (x)+ ) + a,(x)y = g(x) C dx

Use x₁ = y, x₂ = y', and x3 = y". x' = Write the given third order linear equation as an equivalent system of first order equations with initial values. - · 2y"" + (t − (1 + 3t¹)) y = −t y(−2) = 3, y′(−2) = 3, y″(−2) = 2 with initial values x = If you don't get this in 2 tries, you can get a hint. 3 cos (t) y' with 18

Find two linearly independent solutions of 2x²y" - xy + (−4x + 1)y = 0, x > 0 of the form Y₁ = x¹(1+ a₁x + a₂x² + aşx³ + ...) Y2 = : x*²(1+b₁x+b₂x² + b3x³ + ...) where T₁ > T2. Enter 71 = a1 = a2 = az = r2 = b₁ = b₂ = b3 In

6) = (" 9) x' -4 a. Find the most general real-valued solution of the linear system 1 ()- ( x(t) = C1 + C2 y(t)

(25 points) Find two linearly independent solutions of 2x²y" - xy' + (−4x + 1)y=0, x > 0 of the form Y₁ = ·x³¹(1+α₁x + ª₂x² + a³x³ + ...) Y₂ = x¹² (1+b₁x + b₂x² + b3x³ + ...) where r₁ > 1₂. Enter T1 = a1 = A₂ = a3 T2 = b₁ = b₂ = b3 = || || ||

Eliminate the arbitrary constant on the given equation: y = C1r + C2e O (2 - 22) y" + (2 + 2) y - (2« – 2) y O (3r - 42) y" - (a + 4) y (4x - 4) y 0 lin O (2? + 2x) y" + (z2 - 2) y-(2x + 2) y = 0 MIRI O (z – 22) / + ( – 4) / 2ar) 3f + ( 4) -(2)y-0

Find two linearly independent solutions of 2x2y"-xy'+(-6x+1)y=0, x>0 of the form

Find the general solution of the system - ( ) * x'(t)