Method of Undetermined Coefficients

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=f(t)$$

where $a,b,c$ are constants, $a\ne 0$, and $f(t)$ is a given function called the nonhomogenous term The general solution structure is as follows:

$$y(t)=y_p(t)+y_c(t)$$

where $y_c(t)$ are solutions of the homogeneous equation:

$$a{y}_c^{\prime \prime }+b{y}_c^{\prime }+c{y}_c=0$$

where $y_p(t)$ is a particular solution of the nonhomogenous (original) equation:

$$a{y}_p^{\prime \prime }+b{y}_p^{\prime }+c{y}_p=f(t)$$

Reminder: $c{\lambda }^2+b\lambda +c=0$ is the Characteristic Equation for the $y_c$ family of solutions

Approach

Step 1: Find complementary solutions $y_c$ Step 2: Find particular solution $y_p$ Constructing a trial term:

$$f(t)=P_n(t)e^{kt}⟹y_p(t)=t^s{P}_n(t)e^{kt}$$

where $s$ is the number of terms in $y_c(t)$ that have $e^{kt}$, where $P_n(t)$ is a n-degree polynomial If any term in the guess for $y_p$ is a solution to $y_c$, multiply by $t$. In other words, multiply by $t^s$ If there are multiple terms, superposition applies. What that means is you substitute one trial solution terms back into the original equation, at a time, in order to find the undetermined coefficients. Note that trig functions are actually exponentials:

$$\sin (t)=\frac{1}{2i}(e^{it}-e^{-it})\cos (t)=\frac{1}{2}(e^{it}+e^{-it})$$

Also recall

Euler's Identity

$$e^{kit}=\cos (kt)-i\sin (kt)$$Link to original

Plug that term into the original equation to solve for undetermined coefficients. Step 3:

$$y(t)=y_p(t)+y_c(t)$$

Example

$3y^{\prime \prime }+y^{\prime }-2y=10e^{4t},y(0)=-1,y^{\prime }(0)=3$ Step 1: Find complementary solutions $y_c$ $3y_c^{\prime \prime }+y_c^{\prime }-2y_c=0$ $3{\lambda }^2+\lambda -2=0⟹\lambda =-1,\frac{2}{3}⟹y_c=C_1{e}^{-t}+C_2{e}^{2/3t}$ Step 2: Find particular solution $y_p$ $10e^{4t}⟹y_p(t)=K{e}^{4}t$ Plug in: $3(K{e}^{4t}{)}^{\prime \prime }+(K{e}^{4t}{)}^{\prime }-2(K{e}^{4t})=50K{e}^{4t}=10e^{4t}⟹K=\frac{1}{5}$ Step 3:

$$y(t)=y_p(t)=y_c(t)=\frac{1}{5}{e}^{4t}+C_1{e}^{-t}+C_2{e}^{2/3t}$$

Now do the IVP. We have already concluded the method of undetermined coefficients. After doing the IVP you get

$$y(t)=\frac{1}{5}{e}^{4t}-\frac{9}{5}{e}^{-t}+\frac{3}{5}{e}^{2/3t}$$