#24 Measure Your Calibre

Calculus Level 4

$\large{x\dfrac{dy}{dx} = y+ x \tan \left(\frac{y}{x}\right)}$

Given the above and that when $x=1$ , $y= 90^\circ$ , and $x=\frac{1}{2}$ , $y= a$ , where $0^\circ \le a \le 90^\circ$ . Find $a$ in degrees.

For other problems: Check your Calibre

The answer is 15.

1 solution

Chew-Seong Cheong
Mar 18, 2017

$\begin{aligned} x\frac {dy}{dx} & = y + x \tan \frac yx & \small \color{#3D99F6} \text{Dividing both sides by }x \\ \frac {dy}{dx} & = \frac yx + \tan \frac yx & \small \color{#3D99F6} \text{Let }u = \frac yx, \ y = ux, \\ u + x \frac {du}{dx} & = u + \tan u & \small \color{#3D99F6} \frac {dy}{dx} = u + x \frac {du}{dx} \\ x \frac {du}{dx} & = \tan u \\ \frac {du}{\tan u} & = \frac {dx}x \\ \int \frac {\cos u \ du}{\sin u} & = \int \frac {dx}x \\ \int \frac {d \sin u}{\sin u} & = \ln x + C \\ \ln (\sin u) & = \ln x + \ln c \\ \implies \sin \frac yx & = cx & \small \color{#3D99F6} \text{Putting }x=1 \\ \sin 90^\circ & = c \\ \implies c & = 1 \\ \implies \sin \frac yx & = x & \small \color{#3D99F6} \text{Putting }x=\frac 12 \\ \sin 2y & = \frac 12 \\ \implies 2y & = 30^\circ \\ y & = \boxed{15^\circ} \end{aligned}$

How do you make out so complex Latex?

Md Zuhair - 4 years, 2 months ago

I understand LaTex very well and I am very particular on presentation.

Chew-Seong Cheong - 4 years, 2 months ago

#24 Measure Your Calibre

For other problems: Check your Calibre

The answer is 15.

1 solution

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