todai-math

Papers (28)
2025
problems1-3 3 todai-engineering-math 6
2024
problem1 1 problem2 1 problem3 1 todai-engineering-math 6
2023
problem1 1 problem2 1 problem3 1 todai-engineering-math 6
2022
problem1 1 problem2 1 problem3 1 todai-engineering-math__paper1 2 todai-engineering-math__paper2 2 todai-engineering-math__paper3 7
2021
todai-engineering-math__paper1 6 todai-engineering-math__paper2 3 todai-engineering-math__paper3 3
2020
todai-engineering-math 6
2019
todai-engineering-math 6
2018
ist 3 todai-engineering-math 6
2017
ist 3 todai-engineering-math 6
2016
todai-engineering-math 6 translated 3
2015
ist 3
2021 todai-engineering-math__paper1

6 maths questions

QI.1 Differentiating Transcendental Functions Compute derivative of transcendental function View
Find the derivative $\frac{\mathrm{d}y(x)}{\mathrm{d}x}$ of the following real function $y(x)$ defined for $0 < x < 1$: $$y(x) = (\arccos x)^{\log x}$$ where $0 < \arccos x < \pi/2$.
QI.2 Integration with Partial Fractions View
Calculate the following indefinite integral: $$\int \frac{x^{2} + x + 2}{x^{3} - px^{2}} \, dx$$ where $p$ is a real constant.
QI.3 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
Calculate the following definite integral: $$I = \int_{0}^{\sin\theta} \frac{\arctan(\arcsin x)}{\sqrt{1 - x^{2}}} \mathrm{~d}x$$ where $0 < \theta < \pi/2$.
QII.1 Systems of differential equations View
Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants.
Derive the simultaneous ordinary differential equations for complex-valued functions $f(x)$ and $g(x)$, based on the change of variables $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.
QII.2 Systems of differential equations View
Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Let $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.
Show that the value of $|f(x)|^{2} + |g(x)|^{2}$ is independent of $x$, where $|A|$ denotes the absolute value of a complex number $A$.
QII.3 Systems of differential equations View
Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Let $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.
Let $a = 0.8$ and $b = 0.6$. Solve the simultaneous ordinary differential equations derived in Question II.1 using the initial values $f(0) = 1$ and $g(0) = 0$, and obtain $f(x)$ and $g(x)$.