Advanced Math Archive: Questions from July 29, 2022
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Solve: \( y^{\prime \prime \prime}-4 y^{\prime \prime}-5 y^{\prime}=9+5 x, y(0)=0, y^{\prime}(0)=0 \) and \( y^{\prime \prime}(0)=4 \)1 answer -
1) Solve the following differential equation \[ y^{\prime \prime}+2 y^{\prime}+y=2 e^{-x}, \quad y(0)=3, \quad y^{\prime}(0)=-1 \]1 answer -
2) Solve the following differential equation \[ y^{\prime \prime}+y=x^{2}, \quad y(0)=0, \quad y^{\prime}(0)=1 \]1 answer -
7) Solve the following differential equation \[ y^{\prime \prime}-3 y^{\prime}+2 y=2 e^{-t}, \quad y(0)=2, \quad y^{\prime}(0)=-1 \]1 answer -
Solve the following initial value problem \[ \begin{array}{l} \frac{\mathrm{d}^{2}}{\mathrm{~d} x^{2}} y(x)+7\left(\frac{\mathrm{d}}{\mathrm{d} x} y(x)\right)+12 y(x)=-8 \cos (2 x)+14 \sin (2 x) \\ y(1 answer -
\( \frac{\mathrm{d}^{2}}{\mathrm{~d} x^{2}} y(x)-5\left(\frac{\mathrm{d}}{\mathrm{d} x} y(x)\right)+6 y(x)=0 ; \quad y(0)=-5 \) \( y(x)= \)3 answers -
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Solve the equation explicitly for \( y \). \[ \begin{array}{l} y^{\prime \prime}+9 y=10 e^{2 t}, y(0)=-1, y(0)=1 \\ y=-\frac{23}{13} \cos (3 t)-\frac{7}{13} \sin (3 t)+\frac{5}{13} t e^{2 t} \\ y=-\fr3 answers -
Let X a set and D C P(X), D closed under an infinite intersection. Note R the ring generated by D. Also, let \pi the smallest system, D C P(X) such that \pi is closed under the following operations: A
Sea \( X \) un conjunto y \( \mathcal{D} \subset \mathcal{P}(X), \mathcal{D} \) cerrado bajo intersección finita. Denotemos por \( \mathcal{R} \) el anillo generado por \( \mathcal{D} \). Además, se1 answer -
Prove that Borel phi-algebra match with the phi-algebra generated by the open sets in R
Prueba que el \( \sigma \)-álgebra de Borel en \( \mathbb{R} \) coincide con el \( \sigma \)-álgebra generado por los conjuntos abiertos en \( \mathbb{R} \).3 answers -
For a space of measure (X,A,mu), prove that: a) if A,B \in A then ... b) if A,B and .. then mu(A)=mu(B)
Para un espacio de medida \( (X, \mathcal{A}, \mu) \), prueba que: a) si \( A, B \in \mathbf{A} \), entonces \[ \mu(A)+\mu(B)=\mu(A \cup B)+\mu(A \cap B) ; \] b) i \( A, B \in \mathbf{A}, \mathrm{y} \1 answer -
\[ \operatorname{diam} E:=\sup \left\{d_{N}(e, f): e, f \in E\right\} . \] Toma \( \delta>0 \) arbitraria. Para cada \( A \subseteq \mathbb{R}^{N} \) llamemos \( \mathcal{C}_{A, \delta} \) a la colecc0 answers