17Calculus Differential Equations - Variation of Parameters
General Form: \(y'' + p(t)y' + q(t)y = g(t)\) |
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Classification: second-order, linear, inhomogeneous |
Alternate Name For Variation of Parameters: Variation of Constants |
Variation of Parameters is a second order technique to solve nonhomogeneous differential equations. The form of the equations we solve using this technique is
\( y'' + p(t)y' + q(t)y = g(t) \) where \( g(t) \neq 0 \).
Although this technique will work when \( g(t) \) is a standard form, we normally use the method of undetermined coefficients which is easier. However, variation of parameters can be used for a wider range of problems.
The equations we use are
\(W =\begin{vmatrix} y_1(t) & y_2(t) \\ y'_1(t) & y'_2(t) \end{vmatrix}\) |
\(\displaystyle{ u'_1(t) = \frac{ \begin{vmatrix} 0 & y_2(t) \\ g(t) & y'_2(t) \end{vmatrix} }{W}}\) |
\(\displaystyle{ u'_2(t) = \frac{ \begin{vmatrix} y_1(t) & 0 \\ y'_1(t) & g(t) \end{vmatrix} }{W}}\) |
\(y_1(t)\) and \(y_2(t)\) are solutions to the homogeneous equation |
The matrix \(W\) is called the Wronskian , and from the above equations you can tell that we require \(W \neq 0\). After evaluating the above matrices for \(u'_1(t)\) and \(u'_2(t)\), we integrate to obtain \(u_1(t)\) and \(u_2(t)\). The final solution to the differential equation is
\( y (t) = u_1(t) y_1 (t) + u_2(t) y_2 (t) \) |
It will help you understand these if you know where the equations come from. So the next section shows the derivation of the these equations and explains step-by-step why we can use them in this form.
Deriving the Variation of Parameters Equations |
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1. Set Up Equations |
First, we form the homogeneous, \(g(t)=0\), solution
\( y_H (t) = c_1 y_1 (t) + c_2 y_2 (t) \).
However, we replace the constants \( c_1, c_2\) with functions \( u_1 (t), u_2 (t) \). This gives us
\( y (t) = u_1(t) y_1 (t) + u_2(t) y_2 (t) \) [1]
Let's take the derivative with respect to \(t\).
\( y'(t) = u_1(t) y'_1(t) + \color{red}{u'_1(t)y_1(t)} + \) \( u_2(t) y'_2(t) + \color{red}{u'_2(t)y_2(t)} \)
Now we set up an unusual condition; we require
\( u'_1(t)y_1(t) + u'_2(t)y_2(t) = 0 \) |
i.e. we take the \(u\)-derivative terms (red terms) and set the sum equal to zero. This removes those terms from the derivative \( y'(t) \) and we are left with
\( y'(t) = u_1(t) y'_1(t) + u_2(t) y'_2(t) \). Let's take the derivative of this again.
\( y''(t) = u_1(t) y''_1(t) + \color{red}{u'_1(t) y'_1(t)} + \) \( u_2(t) y''_2(t) + \color{red}{u'_2(t) y'_2(t)} \)
Now we set up another condition.
\( u'_1(t) y'_1(t) + u'_2(t) y'_2(t) = g(t) \) |
Again, we take the \(u\)-derivatives terms (red terms) and set the sum equal to \( g(t) \) from the original differential equation.
Finally, we have the two equations
\(\begin{array}{lclcl}
u'_1(t)y_1(t) & + & u'_2(t)y_2(t) & = & 0 \\
u'_1(t) y'_1(t) & + & u'_2(t) y'_2(t) & = & g(t)
\end{array}\)
with the two unknowns \( u'_1(t), u'_2(t) \).
2. Solving For Two Unknowns |
Using Cramer's Rule to solve a system of equations, we have the Wronskian
\(
W =
\begin{vmatrix}
y_1(t) & y_2(t) \\
y'_1(t) & y'_2(t)
\end{vmatrix}
\)
which we use to solve for \( u'_1(t) \) and \( u'_2(t) \)
\(\displaystyle{ u'_1(t) = \frac{ \begin{vmatrix} 0 & y_2(t) \\ g(t) & y'_2(t) \end{vmatrix} }{W} }\) |
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\(\displaystyle{ u'_2(t) = \frac{ \begin{vmatrix} y_1(t) & 0 \\ y'_1(t) & g(t) \end{vmatrix} }{W} }\) |
3. Finishing Up |
Now we integrate the equations for \( u'_1(t)\) and \( u'_2(t) \) and plug the results into equation [1] to get our final answer. If we have initial conditions, we would need to use them to find the constants that result from the integration in the last step.
There are two ways to think about how to write the final solution. You can integrate \( u'_1(t)\) and \( u'_2(t) \) to get
\( u_1(t) + c_1\) and \( u_2(t) + c_2 \) and write the final solution as
\( y (t) = (u_1(t) + c_1) y_1 (t) + \) \( ( u_2(t) + c_2) y_2 (t) \)
or, as the video below shows, you can think of the solution \( y_p (t) = u_1(t) y_1 (t) + u_2(t) y_2 (t) \) as a particular solution to which you add the homogeneous solution to get
\( y (t) = y_p(t) + y_h(t) = \) \( u_1(t) y_1 (t) + u_2(t) y_2 (t) + \) \( c_1y_1(t) + c_2y_2(t) \)
As you can see, these are just two different ways of looking at the same thing. It is best to follow what your instructor requires.
Here is a video showing the same derivation. The presenter does a very good job explaining the equations.
Dr Chris Tisdell - variation of parameters [13mins-40secs]
video by Dr Chris Tisdell |
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Here is another video discussing variation of parameters. This video is quite in-depth and he starts out with some matrix theory that supports the later discussion. This video is included here for completeness but is not necessary for using variation of parameters.
MIT OCW - variation of parameters [46mins-52secs]
video by MIT OCW |
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Practice
Instructions - - Unless otherwise instructed, solve these differential equations using the variation of parameters technique. Give your answers in exact terms and completely factored.
Basic Problems |
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Problem Statement |
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\( y'' - 5y' + 6y = 4e^t \)
Final Answer |
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Problem Statement |
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\( y'' - 5y' + 6y = 4e^t \)
Solution |
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We could use the method of undetermined coefficients to solve this problem more easily. However, we are asked to use variation of parameters. We will work through the solution by going through the derivation of the equations.
First, we solve the homogeneous equation. |
\( y'' - 5y' + 6y = 0 \) |
The characteristic equation is |
\( r^2 - 5r + 6 = 0 \) |
\( (r-3)(r-2) = 0 \) |
So our homogeneous solution is |
\( y_H(t) = c_1e^{2t} + c_2e^{3t} \) |
Now, we work on the particular solution by replacing the constants with functions of t, \( y(t) = u_1(t)e^{2t} + u_2(t)e^{3t} \), and taking the derivative with respect to t.
\(y'(t) = u_1(t)(2e^{2t}) + \color{red}{u'_1(t)e^{2t}} + u_2(t)(3e^{3t}) + \color{red}{u'_2(t)e^{3t} }\)
For our first equation, we take the terms in red and set them equal to zero to get \( u'_1(t)e^{2t} + u'_2(t)e^{3t} = 0\) [ 1 ]
This leaves us with \( y'(t) = u_1(t)(2e^{2t}) + u_2(t)(3e^{3t}) \), which we now take the derivative of with respect to t giving us
\( y''(t) = 2u_1(t)(2e^{2t}) + \color{red}{ 2u'_1(t)e^{2t} } + 3u_2(t)(3e^{3t}) + \color{red}{ 3u'_2(t)e^{3t} } \)
Setting the red terms equal to \( g(t) = 4e^t \), we have
\( 2u'_1(t)e^{2t} + 3u'_2(t)e^{3t} = 4e^t \)
Now, we have the two equations we need, the last one and equation [1].
\( u'_1(t)e^{2t} + u'_2(t)e^{3t} = 0 \) |
\( u'_1(t)(2e^{2t}) + u'_2(t)(3e^{3t}) = 4e^t \) |
Now we need to solve for \( u'_1(t)\) and \( u'_2(t) \).
\(\displaystyle{ W = \begin{vmatrix} e^{2t} & e^{3t} \\ 2e^{2t} & 3e^{3t} \end{vmatrix} }\) |
\( W = 3e^{5t} - 2e^{5t} = e^{5t} \) |
\(\displaystyle{ u'_1(t) = \frac{ \begin{vmatrix} 0 & e^{3t} \\ 4e^t & 3e^{3t} \end{vmatrix} }{W} }\) |
\(\displaystyle{ u'_1(t) = \frac{0 -4e^{4t}}{e^{5t}} = -4e^{-t} }\) |
\(\displaystyle{ u_1(t) = \int{-4e^{-t} dt} = 4e^{-t} + c_1 }\) |
\(\displaystyle{ u'_2(t) = \frac{ \begin{vmatrix} e^{2t} & 0 \\ 2e^{2t} & 4e^t \end{vmatrix} }{W} }\) |
\(\displaystyle{ u'_2(t) = \frac{4e^{3t}-0}{e^{5t}} = 4e^{-2t} }\) |
\( u_2(t) = \int{4e^{-2t} dt} = -2e^{-2t} + c_2 \) |
Substituting these results into |
\( y(t) = u_1(t)e^{2t} + u_2e^{3t} \) |
gives us |
\( y(t) = [ 4e^{-t} + c_1 ] e^{2t} + [ -2e^{-2t} + c_2 ] e^{3t} \) |
\( y(t) = 4e^t+ c_1 e^{2t} - 2e^t + c_2 e^{3t} \) |
\( y(t) = 2e^t + c_1 e^{2t} + c_2 e^{3t} \) |
Final Answer |
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\( y(t) = 2e^t + c_1e^{2t} + c_2e^{3t} \) |
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close solution
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Problem Statement |
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\(y_1=x\) and \(y_2=x^3\) are solutions to the homogeneous equation, find the general solution of \( x^2y'' - 3xy' + 3y = 4x^7 \)
Final Answer |
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Problem Statement |
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\(y_1=x\) and \(y_2=x^3\) are solutions to the homogeneous equation, find the general solution of \( x^2y'' - 3xy' + 3y = 4x^7 \)
Solution |
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606 video
video by PatrickJMT |
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Final Answer |
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\(y = x^7 / 6 + c_1x + c_2x^3\) |
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close solution
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Problem Statement |
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\( 2y'' - y' - y = 2e^t \)
Solution |
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2218 video
video by The Lazy Engineer |
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close solution
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Intermediate Problems |
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Problem Statement |
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\(\displaystyle{ y'' - 2y' + y = \frac{e^x}{x^2+1} }\)
Final Answer |
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Problem Statement |
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\(\displaystyle{ y'' - 2y' + y = \frac{e^x}{x^2+1} }\)
Solution |
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600 video
video by Dr Chris Tisdell |
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Final Answer |
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\(\displaystyle{ y = Ae^x + Bxe^x - \frac{e^x}{2}\ln(x^2+1) + xe^x\arctan(x) }\) |
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close solution
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Problem Statement |
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\( y'' + y = \tan(x) \)
Final Answer |
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Problem Statement |
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\( y'' + y = \tan(x) \)
Solution |
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Here are two video solutions by too different instructors. If, after watching the first video, you need more explanation, perhaps the second one will help.
601 video
video by Dr Chris Tisdell |
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601 video
video by The Lazy Engineer |
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Final Answer |
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\( y = -\cos(x)\ln\abs{\sec(x) + \tan(x)} + A\cos(x) + B\sin(x) \) |
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close solution
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Problem Statement |
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\( y'' + 4y = \csc(2x) \)
Final Answer |
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Problem Statement |
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\( y'' + 4y = \csc(2x) \)
Solution |
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603 video
video by Dr Chris Tisdell |
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Final Answer |
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\(\displaystyle{ y = A\cos(2x) + B\sin(2x) - \frac{x}{2}\cos(2x) + \frac{1}{4} \sin(2x) \ln\abs{\sin(2x)} }\) |
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close solution
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Problem Statement |
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\(\displaystyle{ y'' + y = \frac{1}{\cos(x)} }\)
Final Answer |
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Problem Statement |
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\(\displaystyle{ y'' + y = \frac{1}{\cos(x)} }\)
Solution |
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604 video
video by Dr Chris Tisdell |
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Final Answer |
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\( y = A\cos(x) + B\sin(x) + \cos(x)\ln\abs{\cos(x)} + x\sin(x) \) |
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close solution
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Problem Statement |
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\(\displaystyle{ y'' - 2y' + y = \frac{e^x}{x^4} }\)
Final Answer |
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Problem Statement |
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\(\displaystyle{ y'' - 2y' + y = \frac{e^x}{x^4} }\)
Solution |
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607 video
video by Krista King Math |
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Final Answer |
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\(\displaystyle{ y = c_1e^x + c_2xe^x + \frac{e^x}{6x^2} }\) |
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close solution
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Advanced Problems |
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Problem Statement |
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\( y'' + 9y = 2\tan(3x) \)
Final Answer |
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Problem Statement |
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\( y'' + 9y = 2\tan(3x) \)
Solution |
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602 video
video by Dr Chris Tisdell |
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Final Answer |
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\(\displaystyle{ y = \frac{-2}{9}\cos(3x)\ln\abs{\sec(3x)+\tan(3x)} + A\cos(3x) + B\sin(3x) }\) |
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close solution
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Problem Statement |
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\( y'' + 3y' + 2y = 5\cos(t) \)
Final Answer |
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Problem Statement |
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\( y'' + 3y' + 2y = 5\cos(t) \)
Solution |
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605 video
video by Dr Chris Tisdell |
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Final Answer |
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\( y(t) = (1/2)e^{-2t} - (1/2)e^{-t} + (1/2)\cos(t) + (3/2)\sin(t) \) |
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close solution
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