Question

#1) Show that the D.E. is exact and solve. $(e^{\theta} - r\sin(\theta)) d\theta + (\cos(\theta))dr = 0$ #2) Show that $2xyy' = 4x^2 + 3y^2$ is a homogeneous type differential equation, and then solve. #3) Given the D.E. $(D)^{4}(D^{2} + 9)^{2}(D + 6)(y) = g(x)$ a) Find the homogeneous solution. b) Find the particular solution if $g(x) = 2x^{2}e^{-6x} + 5 + 11e^{-6x}$ c) Find the particular solution if $g(x) = x\cos2x + \sin3x$ #4) Find a general solution to the D.E. using variation of parameters: $y'' + 16y = \sec (4t)$. #5) Find the Laplace Transform of $f(t)$ using the definition: $f(t) = \begin{cases} e^{2t}, & 0<t<3 \\ 1, & 3<t \end{cases}$

          #1) Show that the D.E. is exact and solve.
$(e^{\theta} - r\sin(\theta)) d\theta + (\cos(\theta))dr = 0$
#2) Show that $2xyy' = 4x^2 + 3y^2$ is a homogeneous type differential equation, and then solve.
#3) Given the D.E. $(D)^{4}(D^{2} + 9)^{2}(D + 6)(y) = g(x)$
a) Find the homogeneous solution.
b) Find the particular solution if $g(x) = 2x^{2}e^{-6x} + 5 + 11e^{-6x}$
c) Find the particular solution if $g(x) = x\cos2x + \sin3x$
#4) Find a general solution to the D.E. using variation of parameters: $y'' + 16y = \sec (4t)$.
#5) Find the Laplace Transform of $f(t)$ using the definition: $f(t) = \begin{cases} e^{2t}, & 0<t<3 \\ 1, & 3<t \end{cases}$

1 show that the de is exact and solve etheta rsintheta dtheta costhetadr 0 2 show that 2xyy 4x2 3y2 is a homogeneous type differential equation and then solve 3 given the de d4d2 92d 6y gx a 18704

Added by Martin M.

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