3.5 non homogeneous equations and method of undetermined coefficients

Non homogeneous

• What does this mean

  • when the right hand side (the equal sign) is not equal to zero

  • $y^{\prime \prime }+p(t)y^{\prime }+q(t)y=g(t)$

• how do we solve

  • We just solve it like a homogeneous equation and we just ignore the right hand side, then we add a $Y_p(t)$ (particular solution)

  • this is called the “method of undetermined coefficients” (MoUC) for example $y^{\prime \prime }+y^{\prime }-2y=e^{3t}$

    • Step 1: find the general solution of

      • $y^{\prime \prime }+y^{\prime }+2y=0$
        • $y_c(t)=c_1{e}^{-2t}+c_2{e}^t$

      • This is called the complementary solution

    • Step 2: find the particular solution $Y_p$ by “guessing”, this is the right hand side in this case it is $e^{3t}$ so we would

      • $\text{guess }{Y}_p(t)=A{e}^{3t},Y_p^{\prime }(t)=3A{e}^{3t},Y_p^{\prime \prime }(t)=9A{e}^{3t}$, plugging this into the ode
        • $9A{e}^{3t}+3A{e}^{3t}-2A{e}^{3t}=e^{3t}⟹A=\frac{1}{10}$

    • Step 3 combine the solutions like $y(t)=Y_c(t)+Y_p(t)$

      • $y(t)=(c_1{e}^{-2t}+c_2{e}^2)+\frac{1}{10}{e}^{3t}$

• General RHS $Y_p$ guys for A,B

  • Polynomial:

    • $Y_p(t)=t^s(A_n{t}^n)+A_{n-1}{t}^{n-1}\dots A_0{t}^0$

      • S is the number of 3.4 Repeated Roots in the r function solution that specificly includes r=0, SO ITS HOW MANY TIMES r=0 SHOWS UP

  • Polynomial times exponential $e^{\alpha t}(a_n{t}^n\dots )$

    • $Y_p(t)=t^s{e}^{\alpha t}(A_n{t}^n\dots )$
      • S is the number of the root $r=\alpha$ that shows up

  • Polynomial times exponential times sin/cos $e^{\alpha t}\cos \beta t(a_n{t}^n\dots )$

    • $Y_p(t)=t^s{e}^{\alpha t}\cos \beta t(A_n{t}^n\dots )+t^s{e}^{\alpha t}\sin \beta t(B_n{t}^n\dots )$
      • s is the number of $r=\alpha +i\beta$ shows up!