Variation of Parameters for Linear ODEs: First- and Higher-Order Methods

All flashcards

What is an inhomogeneous linear differential equation?

A linear differential equation with a forcing term that is not identically zero, such as $y''+y=F(t)$.

How is the general solution of a linear inhomogeneous equation written?

As $y=y_h+y_p$, where $y_h$ is the general homogeneous solution and $y_p$ is one particular solution.

What is the key substitution in variation of parameters?

Replace the constant coefficients in a linear combination of homogeneous solutions with functions of the independent variable.

Why is variation of parameters considered a general method?

It constructs a particular solution systematically from homogeneous solutions without requiring a guessed solution form.

What is Duhamel's principle related to?

A related approach for certain inhomogeneous linear evolution equations, including heat and wave equations.

Quiz questions

1. For a second-order equation, variation of parameters uses $u_p=A(x)u_1(x)+B(x)u_2(x)$. What is the purpose of imposing $A'u_1+B'u_2=0$?Answer: To simplify the first derivative of the proposed particular solution.
2. For $u''+p(x)u'+q(x)u=f(x)$, let $W=u_1u_2'-u_2u_1'$. Which formulas give the coefficient derivatives in variation of parameters?Answer: $A'=-u_2f/W$ and $B'=u_1f/W$
3. For $x''(t)+x(t)=F(t)$ with $x(0)=x'(0)=0$, which integral gives the response to the forcing?Answer: $x(t)=\int_0^t F(s)\sin(t-s)\,ds$
4. For $y'+p(x)y=q(x)$, set $y_p=C(x)e^{-\int p(x)\,dx}$. What equation for $C'$ follows after substitution?Answer: $C'=q(x)e^{\int p(x)\,dx}$
5. When integrating the coefficient derivatives in variation of parameters, why can the integration constants be omitted from a particular solution?Answer: They add only homogeneous solutions, which are included in the general solution.