Q: Find the asymptotes of $y^2(x - b) = x^3 + a^3$.

By exploiting the fundamental symmetric properties of odd and even functions during integration over $[-\pi, \pi]$, the Fourier cosine coefficient accurately evaluates to $a_n = \frac{4(-1)^n}{n^2}$.

Q: Provide order and classification of the partial differential equation:

Cauchy's strict mathematical definition states that a function $f(x)$ is continuous at $c$ if, for any $\epsilon > 0$, there exists a $\delta > 0$ such that $|x - c| < \delta$ guarantees $|f(x) - f(c)| < \epsilon$.

Q: If $y_1 = x^2$ is one of the independent solutions of the differential equati…

Euler's fundamental theorem states that for any continuously differentiable homogeneous function $u(x,y)$ of exact algebraic degree $n$, the strict partial differential relation $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u$ holds true.

Q: Solve the initial value problem: $(1 + y) \, dx + (x + 2y) \, dy = 0, y(0) = 1$.

By applying the substitution $2x = t$, this improper definite integral perfectly transforms into the standard Gamma function $\Gamma(7)$, rigorously evaluating to the exact rational fraction $\frac{45}{8}$.

Q: Solve $\frac{dy}{dx} = \cos(x + y) + \sin(x + y)$.

By carefully setting up the iterated double integral over the triangular planar region bounded by $x+y=1$ in the first quadrant, the exact mathematical volume evaluates to $1/24$.

Q: Write a short note on the symmetry of Cartesian curves.

Reversing the rigid order of integration physically splits the complex triangular region horizontally, resulting in the sum of two distinct mathematical integrals: $\int_0^a \int_0^y f(x,y) \, dx \, dy + \int_a^{2a} \int_0^{2a-y} f(x,y) \, dx \, dy$.

Question 1

What is the range of $x$ for which the function $y = x^4 - 6x^3 + 12x^2 + 5x …

Accepted Answer

By factoring out the highest power of $n$ from both the numerator and the denominator, the limit fundamentally evaluates exactly to the rational fraction $5/7$.

Question 2

Find the radius of curvature at the origin for the curve $x^3 + y^3 - x^2 + y…

Accepted Answer

The infinite power series expansion for the logarithmic function $\log(1+x)$ is rigorously defined as $x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$, which absolutely converges for $-1 < x \le 1$.

Question 3

Find the asymptotes of $y^2(x - b) = x^3 + a^3$.

Accepted Answer

By exploiting the fundamental symmetric properties of odd and even functions during integration over $[-\pi, \pi]$, the Fourier cosine coefficient accurately evaluates to $a_n = \frac{4(-1)^n}{n^2}$.

Question 4

Provide order and classification of the partial differential equation:

Accepted Answer

Cauchy's strict mathematical definition states that a function $f(x)$ is continuous at $c$ if, for any $\epsilon > 0$, there exists a $\delta > 0$ such that $|x - c| < \delta$ guarantees $|f(x) - f(c)| < \epsilon$.

Question 5

If $y_1 = x^2$ is one of the independent solutions of the differential equati…

Accepted Answer

Euler's fundamental theorem states that for any continuously differentiable homogeneous function $u(x,y)$ of exact algebraic degree $n$, the strict partial differential relation $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u$ holds true.

Question 6

Solve the initial value problem: $(1 + y) \, dx + (x + 2y) \, dy = 0, y(0) = 1$.

Accepted Answer

By applying the substitution $2x = t$, this improper definite integral perfectly transforms into the standard Gamma function $\Gamma(7)$, rigorously evaluating to the exact rational fraction $\frac{45}{8}$.

Question 7

Solve $\frac{dy}{dx} = \cos(x + y) + \sin(x + y)$.

Accepted Answer

By carefully setting up the iterated double integral over the triangular planar region bounded by $x+y=1$ in the first quadrant, the exact mathematical volume evaluates to $1/24$.

Question 8

Write a short note on the symmetry of Cartesian curves.

Accepted Answer

Reversing the rigid order of integration physically splits the complex triangular region horizontally, resulting in the sum of two distinct mathematical integrals: $\int_0^a \int_0^y f(x,y) \, dx \, dy + \int_a^{2a} \int_0^{2a-y} f(x,y) \, dx \, dy$.

Question 9

Solve $(D^4 + a^4)y = 0, a > 0$.

Accepted Answer

By mathematically computing the partial derivatives of each vector component and evaluating them exactly at the coordinate point $(3, -1, -2)$, the scalar divergence is rigorously determined to be $-3$.

Question 10

If $u = (x^2 + y^2 + z^2)^{-1/2}$, show that $\frac{\partial^2u}{\partial x^2…

Accepted Answer

Stokes' Theorem establishes a profound mathematical equivalence between the macroscopic circulation (line integral) of a vector field around a closed curve and the microscopic rotation (surface integral of the curl) over any bounding surface.

Question 11

If $	heta = t^n \exp(-r^2/4t)$, find the value of $n$ for which the relation

Accepted Answer

This fundamental mathematical proof establishes the exact algebraic relationship between the Beta and Gamma functions by representing their product as a double integral in Cartesian coordinates, and then elegantly transforming it into polar coordinates to isolate the component functions.

Question 12

Trace the polar curve $r = 3 + 2\cos 	heta$.

Accepted Answer

The convergence of the infinite mathematical series is rigorously tested using Raabe's Test. After determining the limit evaluates exactly to $1/2$ (which is strictly less than 1), the mathematical sequence is unequivocally classified as a divergent series.

Question 13

Find the equation of the cubic which has the same asymptotes as the curve $x^…

Accepted Answer

By determining the exact Fourier series for the even parabolic function $f(x) = x^2$ and cleverly evaluating the resulting infinite trigonometric series at $x=0$, we successfully deduce the elegant mathematical identity $\frac{\pi^2}{12} = \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \dots$

Question 14

If $u = \sin^{-1}\left(\frac{x+y}{\sqrt{x}+\sqrt{y}}\right)$, show that $x\fr…

Accepted Answer

By rigorously calculating the 3D gradient vector $
abla F$ at the specified coordinate point, we definitively derive the tangent plane equation $x - 3y - 2z + 7 = 0$ and the orthogonal normal line equation $\frac{x-1}{1} = \frac{y-2}{-3} = \frac{z+1}{-2}$.

Question 15

Find the solution of the differential equation: $\frac{d^3y}{dx^3} - 9\frac{d…

Accepted Answer

By strictly solving the system of first-order partial derivatives and verifying with the second derivative test, the absolute maximum of the function $f(x,y)$ is mathematically proven to occur at $(1/2, 1/3)$ with a value of $1/432$.

Question 16

Solve the differential equation: $(x+a)^2 \frac{d^2y}{dx^2} - 4(x+a) \frac{dy…

Accepted Answer

By radically transforming the complex Cartesian double integral into polar coordinates, the integration becomes mathematically viable, cleanly evaluating step-by-step to the exact logarithmic value $\frac{1}{4} \ln(\sqrt{2} + 1)$.

Question 17

Reduce the partial differential equation $y + 2zq = q(4xp + yq)$ to Clairaut'…

Accepted Answer

This vector calculus identity is rigorously proven by meticulously applying the divergence operator $
abla \cdot$ to the cross product using strict vector product rules and exploiting the invariant cyclic properties of the scalar triple product.

Question 18

Find the minimum value of the function $x + y + z$ subject to the condition $…

Accepted Answer

This advanced integral calculus problem strictly determines the exact volumetric capacity of a solid spindle generated by revolving a hypocycloid (astroid) perfectly around the x-axis. Using rigorous parametric substitution and Beta function integration techniques, the final derived volume is mathematically proven to be $\frac{32\pi a^3}{105}$.

Question 19

Solve the differential equation: $\cos x \frac{d^2y}{dx^2} + \sin x \frac{dy}…

Accepted Answer

This problem meticulously proves two specific identities related to a complex logarithmic function. By defining the sum of partial derivatives as a differential operator $D$, we systematically compute $D(u) = \frac{3}{x+y+z}$ and rigidly apply the operator a second time to flawlessly derive $D^2(u) = \frac{-9}{(x+y+z)^2}$.

Question 20

Show that the coordinates $(\alpha, \beta)$ of the center of curvature of the…

Accepted Answer

This complex problem requires deriving the half-range Fourier cosine series for the linear function $f(x) = 2x - 1$ strictly over the interval $(0, 1)$. By evaluating the rigorously constructed infinite series at the boundary $x=0$, we mathematically deduce the specific infinite sum identity $\frac{\pi^2}{8} = 1 + \frac{1}{9} + \frac{1}{25} + \dots$

Question 21

Solve by method of variation of parameters: $\frac{d^2y}{dx^2} + (1 - \cot x)…

Accepted Answer

This highly geometric problem elegantly determines the total physical volume of a 3D tetrahedron bounded by Cartesian planes by employing an affine coordinate transformation. This rigorous mapping translates the complex skewed tetrahedron perfectly into a standard unit simplex, resulting in a finalized volume of $\frac{abc}{6}$.

Question 22

Write Charpit's equations. Find a general solution of $p^2x + q^2y = u, u = u…

Accepted Answer

This highly rigorous proof verifies the Gauss Divergence Theorem by explicitly calculating both the 3D volume integral of the field's divergence and the cumulative 2D surface integrals of its total flux across all four faces of a unit tetrahedron. Both completely independent methods evaluate precisely to $\frac{1}{8}$.

Mathematics I