Solve the second order differential equation d^2y/dx^2 - 4dy/dx + 5y = 15cos(x), given that when x = 0, y = 1 and when x = 0, dy/dx = 0

As we know, a general solution for a given differential equation is the complimentary solution + the particular solution/integral, this case is no different. To solve for the complimentary solution, form the auxillary equation by ignoring all non-y terms, in this case the 15cos(x), and setting the LHS to equal 0. Define the variable K to be the variable for our auxillary equation and the DE becomes K^2 -4K + 5 = 0. Solving the quadratic yields roots 2+i and 2-i. We know that complimentary solutions appear in the form Ae^lx + Be^mx where l and m are the roots of the aux. equation, however by application of De Moivre's Theorem by splitting the exponents, and grouping terms, you'll see that the comp. solution appears as e^2x(Acos(x) + Bsin(x)), where A and B are constants to be found.
For the particular integral, as the RHS in the form of cos(x), let the particular solution y = ecos(x) + fsin(x), where e and f are also constants to be found. Differentiating once yields us fcos(x) - esin(x), and differentiating once more yields -ecos(x) - fsin(x). By substituting these in to the left hand side, we can then compare coefficients and we find that e and f are 15/8 and -15/8 respectively. Adding the complimentary solution to the particular integral gives us e^2x(Acos(x) + Bsin(x)) +15/8(cos(x)-sin(x)). Considering our boundary conditions, we can set the RHS to 1 and substitute x = 0 on the left to get 1 = A + 15/8, getting A = 7/8. By differentiating once, applying the product rule and chain rule we get 2e^2x(Acos(x) + Bsin(x)) + e^2x(Bcos(x) - Asin(x)) - 15/8cos(x) -15/8sin(x). By setting the RHS to 0 and substituting x = 0 on the LHS. We obtain 0 = 2A + B - 15/8. Knowing that A = -7/8 and solving for B, we get B = 29/8. Thus, by grouping terms to tidy up, we obtain the final solution of y = e^2x/8(29sin(x) - 7cos(x)) + 1/8(cos(x) - sin(x)).

AG
Answered by Aryan G. Further Mathematics tutor

2997 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

How would go about finding the set of values of x for which x+4 > 4 / (x+1)?


I don't understand how proof by mathematical induction works, can you help?


I don't know what I am doing when I solve differential equations using the integrating factor and why does this give us the solutions it does?


How do you find the derivative of arcsinx?


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences