AdvancedMathematicsHigh School (11–12)Functions and CalculusNumerical

A model of a cooling process, written in plain scaled numbers x and y > 0, obeys the equation y dx…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A model of a cooling process, written in plain scaled numbers $x$ and $y > 0$ , obeys the equation $y\,dx + (2x - \ln y)\,dy = 0$ . A student notices that it is not exact. The quantity $\dfrac{N_x - M_y}{M}$ depends on $y$ alone, so a factor in $y$ makes the equation exact. After the factor is used the solution is $xy^2 - \left(\dfrac{y^2}{2}\ln y - \dfrac{y^2}{4}\right) = C$ . The solution passes through the point $(1, 1)$ . Find the constant $C$ , the value of $x$ where $y = 2$ , and the value of $x$ where $y = e$ . Give $x$ to four decimal places.

Need a reminder? The key ideaShow

$M\,dx + N\,dy = 0$ is exact when $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$ , and then $F(x, y) = C$ with $F_x = M$ and $F_y = N$ . If $\dfrac{M_y - N_x}{N}$ depends on $x$ only, $\mu = e^{\int \frac{M_y - N_x}{N}dx}$ . If $\dfrac{N_x - M_y}{M}$ depends on $y$ only, $\mu = e^{\int \frac{N_x - M_y}{M}dy}$ . For homogeneous $M$ , $N$ use $\mu = \dfrac{1}{Mx + Ny}$ .

The worked solution

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