Help with Linear Algebra and Differential Equation!! Could anyone show me a thor

Question

Help with Linear Algebra and Differential Equation!! Could anyone show me a thoroughly worked-out solution?

Check that y(x) = 0 is a solution of the differential equation y" + 3y' + 2y = 0 which satisfies the initial conditions y(0) = 0, y'(0) = 0. Is y(x) = 0 the only solution that satisfies the differential equation and initial conditions? If y(x) =0 is not the only such solution, can you find another one? Does your answer contradict Theorem 4 of Chapter 5.1 which states that we should expect two linearly independent solutions from a linear, second-order, homogeneous, ODE?

Explanation / Answer

We are given a linear homogeneous ode with constant coefficients

So y=exp(kx)

Substituting gives

k^2+3k+2=0

SOlutions to this are:k=-1,-2

So,it has two linearly independent solutions

exp(-x) and exp(-2x)

And any solution with given initial condition si a linear combination of these two solutions

So, y(x)=Aexp(x)+Bexp(-2x)

y(0)=A+B=0

y'(0)=A-2B=0

Hence, A=B=0

So, y(x)=0 is the solution with given initial conditions

This does not contradict the Theorem 4 because we have computed a particular solution. In general any solution will be a linear combination of the two linearly indepdnent solutions

Related Questions

Navigate

• Browse (All)
• Browse H
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.