Find the equation for a linear line, by knowing the coordinates of two points
Question: A line goes through points (-3, 9) and (2, 4). Find the equation of the linear line. Answer: There […]
Category: Mathematics
Inequality relationship between trigonometric functions
Question: (1) The range of \(x\) is: \(0^o \leq x \leq 180^o\). Solve \(\cos^4 x > \sin^4 x\). (2) The […]
How to solve the differential equation: \((x^2\cos x – y)dx + xdy = 0\)
Question: Solve \((x^2\cos x – y)dx + xdy = 0\). Answer: Rewrite the equation as follows: \[ \frac{dy}{dx}-\frac{y}{x}=-x\cos x \] […]
Solve the differential equation: \(x\frac{dy}{dx}+y=x\ln x\)
Question: (1) Solve \(x\frac{dy}{dx}+y=0\) (2) Solve \(x\frac{dy}{dx}+y=x\ln x\) by using the result from (1). Answer: (1) The original differential equation […]
Solve the differential equation, \(\frac{dy}{dx}+P(x)y=Q(x)\).
Question: (1) Find the general solution of \(\frac{dy}{dx}+P(x)y=Q(x)\), where \(P(x)\) and \(Q(x)\) are functions of \(x\). (2) Solve the following […]
Finding the enclosed area between a parabola and a circle
Question: A parabola is given as \[ y=\frac{1}{4}x^2-1 \] A circle is also given in the same plane: \[ x^2+y^2=16 […]
Infinite integral of the exponential and trigonometric function
Question: Find the value of the following integral: \[ \int^{\infty}_{-\infty}e^{-x^2}\cos 2bxdx \] Answer: Let \[ \int^{\infty}_{-\infty}f(x,b)dx = 2\int^{\infty}_{0}f(x,b)dx \] where […]
Solving for the eigenvalue and eigenvector of a matrix
Question: (1) Find the eigenvalues and eigenvectors of a matrix, \[ A=\left(\begin{array}{c}-5 & 6\\-4 & 5\end{array}\right) \] (2) If \(C\) […]
Conditions for the regular function: Complex variables
Question: Consider a complex function, \(f(z)=u(x,y)+iv(x,y)\). When \(u(x,y)=\frac{x}{x^2+y^2-2y+1}\), find the \(f(z)\) which can be a regular function. Answer: According to […]
Find a volume: Triple integral
Question: Evaluate \(\iiint_D x\sqrt{1+x^2+y^2+z^2}dxdydz\) The domain is \(D=\{(x,y,z) | x\geq 0, y\geq 0, x^2+y^2+z^2\leq 1 \}\). Answer: Use the polar […]