Method of undertimed coefficents video

Method of Undetermined coefficents is a method to solve non-homogenous differential equations.

General solution is in the form below

where the second order non homogenous ODE has the form

• $y_h$ is the solution to the homogenous ODE $a(x)y''+b(x)y'+c(x)y=0$
• $y_p$​​​, the particular solution, is a function that satisfies the non-homogenous equation. This can be determined by using the method of undetermined coefficents.

Problem 5: Find the General solution the non homogenous ODE $y''-8y'+16y=16e^{6t}$​

Solution 5:

Solve for $y_c$ first.

So, $y_c=c_1e^{4t} + c_2e^{4t}$​​

Use method of undetermined coefficents

This simplifies to

So, $y_p=4e^{6t}$​

General solution is $y = c_1e^{4t} + c_2e^{4t} + 4e^{6t}$​

Problem 6: Find intial values and solve the DE: $y'' + 25y=e^{-t}$​​​, $y(0)=y_0$​​​, $y'(0)=y'_0$​​​​​​. Suppose we know that $y(t) \rightarrow 0$​ as $t \rightarrow \infin$​.

Solution 6:

Get general solution first $y = y_h + y_p$​. For $y_h$, $m=\pm5i$ since characteristic equation is $m^2+25=0$​. So, $y_h$ is $c_1cos(5t) + c_2sin(5t)$​.

Solve for $y_p$​​​ next. $y_p = Ae^{-t}$​​​. Plug in to the equation: $Ae^{-t} + 25Ae^{-t} = e^{-t}$​​​. Simplifies to $26Ae^{-t}=e^{-t}$​​​, making $A=\frac{1}{26}$​​

General solution then is $y = c_1cos(5t) + c_2sin(5t) + \frac{1}{26}e^{-t}$​.

Find initial conditions. First, we do $y(0)$​​. $y(0) = c_1 + 1/26$​​​​​, where $c_1$​​ and $c_2$​​ have to be 0 here in order for $\lim_{x \to \infty} y=0$​​​​​, and so $y(0) = 1/26$​​​. We get that limit because we know that $y(t) \rightarrow 0$ as $t \rightarrow \infty$. That is our first initial condition. Now, our second initial condition $y'(0) = -c_1sin(5*0) + c_2cos(5*0)+1/26=c_2-1/26$ and since $c_2$ is also 0, $y'(0)=-1/26$. Our initial conditions are $y(0)=1/26$, and $y'(0)=-1/26$​​​.

Now, solve for y given the initial conditions. $y=\frac{1}{26}e^{-t}$​ from plugging in both the initial conditions and solving like a normal IVP.

Answers: $y(0)=1/26$​, $y’(0)=-1/26$​, and $y=\frac{1}{26}e^{-t}$​​​