Homogeneous Linear Equations


Part d: Reduction of order



Remember

Each differential equation has one (non-trivial) homogeneous solution given. Use it to make a useful substitution and solve for the substituted variable, allowing you to unlock the full solution set.


For example, an eqaution like

p1(t)y′′+p2(t)y′+p3(t)y=0p_1(t)y'' + p_2(t)y' + p_3(t)y = 0

can be replaced with an equation

q1(t)v′′+q2(t)v′=0.q_1(t)v'' +q_2(t)v' = 0.

Because there is no vv term, you can just use integrating factor to solve the equation for v′v', then integrate to find vv, and you have the full solution set for yy.



You must meet all initial conditions for the answer to be complete.

Please enter any multiplication explicitly with * symbols so the parcer can understand your answer, like this:

2t3e5t2⟶2∗t3∗e(5∗t2).2t^3e^{5t^2} \longrightarrow 2*t^3*e^{(5*t^2)}.


Problem 1

Find a family of functions that solves the equation and satisfies the conditions below with t>0. e^t is one solution.

ty′′−(1+t)y′+y=0ty'' - (1 + t)y' + y = 0

Initial Conditions to Check:

  • ⬜️y(1) = 1 and y'(1) = 0
  • ⬜️y(1) = 0 and y'(1) = 1