
Differential Equations Ordinary and Partial Differential Equations Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 19th, 2017, 05:08 AM  #1 
Newbie Joined: Feb 2017 From: Karachi,Pakistan Posts: 1 Thanks: 0  Linear ordinary differential equation
Here is an equation Vsinwt=iR+Ldi/dt i have to solve this equation with linear method ,i.e i'+p(t)i=Q(t) i have solved it to the step Li'e^rt+iRe^rt=e^rtVsinwt by initial steps and by finding integrating factor ,but now i stuck ,just don't know that what to do with that L with first most term,because if there is no L,there is a formula of d/dx(u.v) on L.h.s and it will proceed for solution of equation. .so can you plz tell me that what to do with that L in the step where i reached or from the main equation given? 
February 19th, 2017, 01:24 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,655 Thanks: 841 
You won't solve this using an integrating factor. You need to first solve the homogeneous equation $R i + L i^\prime = 0$ and then use the method of undetermined coefficients to solve for the particular solution given the driving function $V \sin(\omega t)$ The final solution is the sum of the homogeneous solution and the particular solution. With a sinusoidal driving function you should try a function of the form $i_p(t) = A \cos(\omega t) + B \sin(\omega t)$ plug that into the differential equation and solve for the $A, ~B$ that makes that equal to the driving function $V \sin(\omega t)$ It's a bit of differentiation and a bunch of algebra. I have confidence you can do it! 
February 19th, 2017, 03:45 PM  #3 
Member Joined: Oct 2016 From: Melbourne Posts: 77 Thanks: 35 
You CAN use the integrating factor method, the IV is "t" and the DV is "i" and all other letters are constants. $\displaystyle \begin{align*} V\sin{ \left( \omega \, t \right) } &= i\,R + L\,\frac{\mathrm{d}i}{\mathrm{d}t} \\ \frac{\mathrm{d}i}{\mathrm{d}t} + \frac{R}{L}\,i &= \frac{V}{L}\,\sin{ \left( \omega\,t \right) } \end{align*}$ The integrating factor is $\displaystyle \begin{align*} \mathrm{e}^{\int{ \frac{R}{L}\,\mathrm{d}t }} = \mathrm{e}^{\frac{R}{L}\,t} \end{align*}$ so multiplying both sides of the equation by this gives $\displaystyle \begin{align*} \mathrm{e}^{\frac{R}{L}\,t}\,\frac{\mathrm{d}i}{ \mathrm{d} t} + \frac{R}{L}\,\mathrm{e}^{\frac{R}{L}\,t}\,i &= \frac{V}{L}\,\mathrm{e}^{\frac{R}{L}\,t}\,\sin{ \left( \omega\,t \right) } \\ \frac{\mathrm{d}}{\mathrm{d}t}\,\left( \mathrm{e}^{\frac{R}{L}\,t}\,i \right) &= \frac{V}{L}\,\mathrm{e}^{\frac{R}{L}\,t}\,\sin{ \left( \omega\,t \right) } \\ \mathrm{e}^{\frac{R}{L}\,t}\,i &= \int{ \frac{V}{L}\,\mathrm{e}^{\frac{R}{L}\,t}\,\sin{ \left( \omega\,t \right) }\,\mathrm{d}t } \end{align*}$ You can now perform the integration using integration by parts twice. 
February 19th, 2017, 05:45 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,091 Thanks: 2360 Math Focus: Mainly analysis and algebra 
The OP has missed that in the form $$i' + p(t)i = Q(t)$$ the coefficient of $i'$ is 1. This essential for the method of the integrating factor to work. 

Tags 
differential, equation, linear, ordinary 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Linear Ordinary Differential Equation with Constant Coefficients  zylo  Calculus  5  November 30th, 2016 07:56 PM 
Ordinary Differential Equation  Abdus Salam  Differential Equations  1  April 28th, 2015 07:39 PM 
ordinary differential equation  aheed  Differential Equations  3  February 22nd, 2014 08:03 AM 
Ordinary differential equation y' = 2y + 4  Norm850  Differential Equations  2  February 1st, 2012 12:57 AM 
an ordinary differential equation  allison711  Differential Equations  3  February 8th, 2008 09:19 AM 