# Method of undetermined coefficients

### Method of undetermined coefficients

#### Lessons

A non-homogeneous constant coefficient second order linear differential equation is of the form:
$Ay''+By'+Cy=d(x)$

The complementary solution to $Ay''+By'+Cy=0$ is $y_c (x)=c_1 f(x)+c_2 g(x)$

And the particular solution to $Ay''+By'+Cy=d(x)$ is $y_p (x)$

So the full general solution to
$Ay''+By'+Cy=d(x)$

Will be $y(x)=y_c+y_p$

If $y_p1 (x)$ is a particular solution for
$Ay''+By'+Cy=d_1 (x)$

And $y_p2 (x)$ is a particular solution for
$Ay''+By'+Cy=d_2 (x)$

Then $y_p1 (x)+y_p2 (x)$ is a solution to
$Ay''+By'+Cy=d_1 (x)+d_2 (x)$

• 1.
a)
What is the Method of Undetermined Coefficients?

b)
How to find sums of particular solutions

• 2.
Using the Method of Undetermined Coefficients
Find the solution to the following differential equation:

$y''-6y'+5y=2e^{3x}$

With initial values $y(0)=-\frac{1}{2}$, and $y' (0)=-\frac{1}{2}$

• 3.
Find the solution to the following differential equation:

$y''+y'-6y=4\sin(2x)$

With initial values $y(0)=\frac{12}{13}$, and $y' (0)=-\frac{10}{13}$

• 4.
Find the solution to the following differential equation:

$y''+y'-6y=-12x^3+3x^2+\frac{19}{6}$

With initial values $y(0)=3$ and $y' (0)=-1$

• 5.
Find the general solution to the following differential equation:
$y''+2y'+y=xe^{3x}+5\cos(3x)$