3.6 Method of variation of parameters

What is MoVop

• When the method of 3.5 non homogeneous equations and method of undetermined coefficients fails

  • The ode has varible coefficents (not euler type)

  • the RHS is complex

    • MoVop aims to solve it

• Suppose that $y_1,y_2$ is a function solution set

• where the functions $u_1$ and $u_2$ are obtained using
  • $u_1(t)=-{\int }_{t_0}^t\frac{y_2(s)g(s)}{W(y_1,y_2)(s)}dx$

Example

• $x^2{z}^{\prime \prime }+2x{z}^{\prime }-6z=e^{\tan x}$

  • this looks like an euler type ode so try $z(x)=x^r$

$$\begin{array}{r}{x}^2(r(r-1)x^{r-2})+2x(r{x}^{r-1})-6x^r=0\\ x^r(r(r-1)+2r-6)=0\\ ⟹y_1=x^{-3},y_2=x^2\\ \text{ wronskian test:}\\ w(w^{-3},x^2)=\left[\begin{array}{cc}{x}^{-3}& x^2\\ -3x^4& 2x\end{array}\right]⟹5x^{-2}=!0\\ \text{ y1 and y2 are linearly independent}\\ yc(t)c_1{x}^3+c_2{x}^2\end{array}$$ $$Q(t)=\{\begin{array}{ll}2,000,000+9,901\sin (2t)-99,010\cos (2t)-1,900,990e^{-0.2t}& \text{for }0\le t<7\text{days}\\ 1,527,249\times e^{-0.2(t-7)}& \text{for }t\ge 7\text{days}\end{array}$$