# Section 1.3 – Classification of Differential Equations

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Given the following differential equations, classify each as an ordinary differential equation, partial differential equation, give the order. If the equation is an ordinary differential equation, say whether the equation is linear or non linear.
1. $$\dfrac{d y}{d x}=3y+x^2$$
2. $$5\dfrac{d^4 y}{d x^4}+y=x(x-1)$$
3. $$\dfrac{\partial N}{\partial t}=\dfrac{\partial ^2N}{\partial r^2}+\frac{1}{r}\dfrac{\partial N}{\partial r}+kN$$
4. $$\dfrac{d x}{d t}=x^2-t$$
5. $$(1+y^2)y''+ty'+y=e^t$$​

$\begin{array}{|c|c|c|c|c|} \hline Part& ODE& PDE & Order & Linear/Non Linear \\ \hline (a)&\checkmark & & 1 & Linear\\ \hline (b)&\checkmark&&4&Linear \\ \hline (c)&&\checkmark&2&Linear \\ \hline (d)&\checkmark&&1&Non Linear \\ \hline (e)&\checkmark&&2&Non Linear\\ \hline \end{array}$

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2.
1. Show that $$f(x)=(x^2+Ax+B)e^{-x}$$ is a solution to $y''+2y'+y=2e^{-x}$ for all real numbers $$A$$ and $$B$$.
2. Find a solution that satisfies the initial condition $$y(0)=3$$ and $$y'(0)=1$$.​

1. For verification that $$f(x)$$ is a solution for part (a), see the video.
2. $$y=(x^2+4x+3)e^{-x}$$

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3. Determine for which values of $$r$$ the function $$t^r$$ is a solution of the differential equation
$t^2y''-4ty'+4y=0, \quad t>0$

$$r=1,4$$

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4. For which values of $$r$$ is the function $$(x-1)e^{-rx}$$ a solution to $$y''-6y'+9y=0$$?

$$r=-3$$

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