Determine the particular solution to the equation 2\frac{dv(t)}{dt} + v(t) = 5t\sin(t)


Determine the particular solution to the equation {eq}2\frac{dv(t)}{dt} + v(t) = 5t\sin(t) {/eq}

Particular Solution:

Given a Linear constant coefficient differential equation with a forcing function in RHS we choose particular integral with following rules:

1) If the forcing function is :

{eq}e^{at} {/eq}

Then :

{eq}y_p=Ae^{at} {/eq}

If Forcing function is sinusoid , then :

{eq}y_p=C\sin(wt)+D\cos(wt) {/eq}

Answer and Explanation: 1

The given differential equation is:

{eq}2\frac{dv}{dt}+v=5t\sin t {/eq}

Since the forcing function is multiplication of linear term and sinusoid ,then particular integral is of the form:

{eq}v_p=(At+B)\sin t+(Ct+D)\cos t {/eq}

Differentiating both sides we get:

{eq}v_p'=(A-Ct-D)\sin t+(At+B+C)\cos t\\ 2v_p'=(2A-2Ct-2D)\sin t+(2At+2B+2C)\cos t {/eq}

Now Particular solution should satisfy the given differential equation:

{eq}2v_p'-+v_p=((A-2C)t+B-2A-2D)\sin t+((2A+C)t+2B+2C+D)\cos t=5t\sin t {/eq}

Comparing coefficients we get:

{eq}A-2C=5\\ 2A+C=0\\ 2B+2C+D=0\\ B+2A=2D {/eq}

Solving we get:

{eq}A=1\\ B=\frac{6}{5}\\ C=-2\\ D=\frac{8}{5} {/eq}

Hence the particular solution is:

{eq}v_p=t(\sin t-2\cos t)+\frac{2}{5}\left(3\sin t+4\cos t\right) {/eq}

Learn more about this topic:

Undetermined Coefficients: Method & Examples
Undetermined Coefficients: Method & Examples


Chapter 10 / Lesson 15

The method of undetermined coefficients is used to solve a class of nonhomogeneous second order differential equations. This method makes use of the characteristic equation of the corresponding homogeneous differential equation.

Related to this Question

Explore our homework questions and answers library