Just uploaded the chapter about the Differential Equations to the maths books page.
You will find the basics of Laplace Transform too. Good Luck.
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I have just uploaded the Hyperbolic Functions chapter at the maths books page. I put the PDF files of exams of some famous Japanese Universities
(You can find them just click the text " Entrance Exams of Japanese Universities 2019" on the bottom of the English web page.) Original exams are written in Japanese, then I translated them into English, and two of these universities questions ( Kyoto and Tokyo) I made answers too. (Generally there are no official corrections for Japanese Entrance Exams.) If you need answers for another universities, please request me from the comments area or e-mail me. Since $$1+i = \sqrt2 (\cos \dfrac{\pi}{4} + i \sin \dfrac{\pi}{4})$$ and $$1-i = \sqrt2 (\cos \dfrac{\pi}{4} - i \sin \dfrac{\pi}{4})$$ then $$\begin{split} (1+i)^n + (1-i)^n &= \left( \sqrt2 (\cos \dfrac{\pi}{4} + i \sin \dfrac{\pi}{4}) \right)^n + \left( \sqrt2 (\cos \dfrac{\pi}{4} - i \sin \dfrac{\pi}{4}) \right)^n \\ &= 2^{\frac{n}{2}} ( \cos \dfrac{n \pi}{4} + i \sin \dfrac{n \pi}{4} ) + 2^{\frac{n}{2}} ( \cos \dfrac{n \pi}{4} - i \sin \dfrac{n \pi}{4} ) \\ &= 2^{\frac{n}{2}+1} \cos \dfrac{n \pi}{4} \end{split}$$ For \(\cos \dfrac{n \pi}{4}\) is positive, $n = 8k, \; n = 8k+1$ or $n=8k+7$, where $k$ is a non-negative integer.
Hence the smallest positive integer $n$ is $$n = 71$$ Let $x$ be the distance from the center $O$ of the sphere and the square $B_1B_2B_3B_4$. The condition (I) means: Today I put rest of the problems of entrance exams of Kyoto Univeristy. I am afraid that they are not presented normally on some browser, then I will put PDF file on the main page of this WEB site 'Entrance Exams of Japanese University 2019'. We fix the coordinates system such that $A(v, w), B(0,0)$ and $C(u,0)$ where $u,v$ and $w$ are positive numbers.
Today it is the question 2 of Kyoto University.
Question 2 is the problem about integers. The point of this question is whether you can find that one of the two consecutive interges must be odd and another one must be even. You see that either $n$ or $n+1$ is even.
Now it is the season of the entrance exams of Japanese Universities.
Each university gives their own exams and I will begin the maths exams of Kyoto University for this year, which is presented on the 25th February 2019. It contains 6 main questions for 150 minutes. Today I will give you the first question; the first one is a bit difficult but the second integrations are very standard. (1) Let $x = \cos \theta$, then $0<x<1$ for $0 < \theta < \frac{\pi}{2}$. |
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