1
core
Differential equations
Calculus
Solve the initial value problem
\[ x^2 y'' + xy' - 4y = 0, \quad y(-1) = 2, \quad y'(-1) = 0. \]
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2
core
Differential equations
Calculus
Let
$f(t) = e^{at}$ and $g(t) = e^{bt}$ $(a \neq b)$.
Verify that $\mathcal{L}(f*g) = \mathcal{L}(f)\mathcal{L}(g)$, as implied by the convolution theorem.
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3
core
Differential equations
Calculus
Use Euler's method with $h = 0.1$ to find approximate values for the solution of the initial value problem
\[ y' + 2y = x^3e^{-2x}, y(0) = 1 \]
at $x = 0.1, 0.2, 0.3$.
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4
core
Differential equations
Calculus
Find a fundamental set of Frobenius solutions of
\[ 2x^2(2+x)y'' + 5x^2y' + (1+x)y = 0. \] Give explicit formulas for the coefficients in the solutions.
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5
hard
Multivariable calculus
Calculus
Verify the general formula \(\frac{d}{dx} \int_a^b f(x, y)dy = \int_a^b \frac{\partial f}{\partial x} dy\), for \(f(x, y) = x + y\), \(a = 0\), \(b = 1\).
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6
hard
Parametric equations and vector functions
Calculus
\(x = 4 \cos \phi\), \(y = 1 - \sin \phi\), \(0 \le \phi \le 2\pi\)
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7
core
Parametric equations and vector functions
Calculus
Focus (\text{-}3, 5) and directrix $y = 1$
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8
core
Parametric equations and vector functions
Calculus
\( r = \frac{15}{3 - 2 \cos \theta} \)
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9
hard
Parametric equations and vector functions
Calculus
Region enclosed by $r=3 \sin \theta$
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10
core
Vector spaces
Linear algebra
Evaluating dot products
1. Let \(\vec{u} = \langle 1, 2 \rangle, \vec{v} = \langle 3, -1 \rangle\) in \(\mathbb{R}^2\). Find \(\vec{u} \cdot \vec{v}\).
2. Let \(\vec{x} = \langle 2, -2, 5 \rangle\) and \(\vec{y} = \langle -1, 0, 3 \rangle\) in \(\mathbb{R}^3\). Find \(\vec{x} \cdot \vec{y}\).
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