Extraordinary Differential Equations #1

Calculus Level 3

$\large (y')^2+2xy'-7yy'-14xy=0$

Variable $y$ is dependent on variable $x$ such that the solutions to the differential equation above are:

$\begin{cases} y=c_1 e^{ax} \\ y=c_2-x^b \end{cases}$

where $a$ and $b$ are real constants. Determine the value of $a^b$ .

57 49 55 53 47 51

1 solution

Tom Engelsman
Jan 12, 2017

The above DE can be factored as:

$(y' - 7y)(y' + 2x) = 0 \Rightarrow y' = 7y, -2x.$

and integrating both solutions yields:

$y = c_1 \cdot e^{7x}$ , $y = -x^2 + c_2$ ,

hence $a = 7, b = 2$ , or $a^{b} = 7^{2} = \boxed {49}.$

A lovely sum!

Md Zuhair - 4 years, 4 months ago