A community for students. Sign up today!
Here's the question you clicked on:
 0 viewing

This Question is Closed

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1\[\large y'+y=x^2+2x\] Let's start by using the method of Undetermined Coefficients. Our complimentary solution \(\large y_c\) is given by the solution to this equation.\[\large y'+y=0\] The characteristic equation is,\[\large r+1=0 \qquad \rightarrow \qquad r=1\]Giving us a complimentary solution of, \(\large \qquad y_c=Ce^{x}\)  We will also need to find the particular solution \(\large y_p\). Our particular solution will be of the same form as the right side of our problem. In this case, we have a polynomial of degree 2. So our particular solution will look something like,\[\large y_p=Ax^2+B(2x)+C\]Where A, B and C are unknown constants. We'll take the derivative of our particular solution and then we can plug in \(\large y_p\) and \(\large y_p^{'}\) into the original problem to solve for A, B and C.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Eventually we're trying to get an answer that consists of the complimentary AND particular solution. \(\large \qquad y=y_c+y_p\)

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1\[\large y_p=Ax^2+2Bx+C\]\[\large y_p^{'}=2Ax+2B\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Plugging these into the original equation gives us,\[y'+y=x^2+2x \qquad \rightarrow \qquad \left(2Ax+2B\right)+\left(Ax^2+2Bx+C\right)=x^2+2x\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1From here we can equate like terms.\[2B+C=0\]\[2Ax+2Bx=2x \qquad \rightarrow \qquad 2A+2B=2\]\[Ax^2=x^2 \qquad \rightarrow \qquad A=1\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Does any of this make sense @guess or should i stop? XD

guess
 one year ago
Best ResponseYou've already chosen the best response.1@zepdrix thank you ,but if you can solve it by laplace Be very grateful

guess
 one year ago
Best ResponseYou've already chosen the best response.1@zepdrix yes the particular solution i think that what he want thanks and don't stop please:)

hartnn
 one year ago
Best ResponseYou've already chosen the best response.1u know whats L[dy/dx] =... ?

hartnn
 one year ago
Best ResponseYou've already chosen the best response.1assuming initial conditions =0 L[dy/dx] = s Y(s) where Y(s) is the Laplace Transform of y(t). so, take Laplace on both sides of dy/dx+y=x^2+2x and can you tell me what you get in first step ??

guess
 one year ago
Best ResponseYou've already chosen the best response.1@zepdrix then what ? is b=1/2 ?

guess
 one year ago
Best ResponseYou've already chosen the best response.1@zepdrix @hartnn @nitz this question came at midterm and ididn't solve it ): help me please

Callisto
 one year ago
Best ResponseYou've already chosen the best response.4Can't we use integrating factor to solve it?

guess
 one year ago
Best ResponseYou've already chosen the best response.1@Callisto mm Can you show me?

mukushla
 one year ago
Best ResponseYou've already chosen the best response.0what Callisto said probabely is the best bet :)

Callisto
 one year ago
Best ResponseYou've already chosen the best response.4\[y'+p(x)y=q(x)\] \[\alpha = exp[\int p(x)dx]\]\[y=\frac{1}{\alpha}\int \alpha [q(x)] dx\]

Callisto
 one year ago
Best ResponseYou've already chosen the best response.4in your case p(x)=1, q(x) = x^2+2x

guess
 one year ago
Best ResponseYou've already chosen the best response.1@Callisto it look like will be best bet really !! continue please :)

Callisto
 one year ago
Best ResponseYou've already chosen the best response.4Ha! Several integration by parts there :S \[y'+y=x^2+2x\] \[\alpha = e^{\int 1 dx}=e^x\] \[y=e^{x}\int e^x(x^2+2x)dx\]\[=e^{x}[e^x(x^2+2x)\int e^x(2x+2)dx]\]\[=e^{x}[e^x(x^2+2x)2(e^x(x+1)\int e^xdx)]\]\[=...\]

Callisto
 one year ago
Best ResponseYou've already chosen the best response.4@guess Can you do it from here? Or still need more help?

hartnn
 one year ago
Best ResponseYou've already chosen the best response.1using Laplace \(L[dy/dx]+L[y]=L[x^2]+2L[x] \\sY+Y=2/s^3+2/s \\ Y=2(1+s^2)/[s^3(s+1)] \\ Y=2/x^32/x^2+4/x4/(s+1)\\\text{skipping the partial fractions part for you.Taking Inverse Laplace}\\ y = x^22x+44e^x\) Someone please verify the final answer @zepdrix or @Callisto

hartnn
 one year ago
Best ResponseYou've already chosen the best response.1\(y=e^{x}\int e^x(x^2+2x)dx , after \:\: simplification, y=x^2\)

guess
 one year ago
Best ResponseYou've already chosen the best response.1@hartnn @Callisto yes please continue i got this put i feel it Error exp^( x) {x^2e^x2e^x} and thank alot for all

Callisto
 one year ago
Best ResponseYou've already chosen the best response.4Method of integrating factor: y'+y=x^2+2x \[\alpha =e^x\]\[y=e^{x}\int e^x(x^2+2x)dx\]\[=e^{x}[e^x(x^2+2x)2\int e^x(x+1)]\]\[=e^{x}[e^x(x^2+2x)2e^x(x+1)+2e^x+C]\]\[=e^{x}(e^xx^2+C)\]\[=x^2+Ce^{x}\]

guess
 one year ago
Best ResponseYou've already chosen the best response.1thank you so much @Callisto @hartnn you're very helper :))

guess
 one year ago
Best ResponseYou've already chosen the best response.1by laplace L[ y`]+L[y]=L[x2]+2L[x] S(Y)+Y(S)=2/S^3+2/S^2 (S+1)Y(S)=2(S+1)/S^3 y(S)=2/S^3 y(x)=x^2 @hartnn @Callisto right ??
Ask your own question
Ask a QuestionFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.