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Find a power series solution in powers of x . Show the details.

y" + y' + yx^2 =0
The equation of a simple harmonic motion is given as d^2x/dt^2 + ω^2x =0. where the symbols have their usual meaning. The dimension of the quantity ω^2 is
a. L^−1
b. M
c. T^−2
d. LT^−2
d^2x/dt^2+2 dx/dt+2x = 4cost +2sint x(0) =0; x(0)=3
Solve this IVP Uxx=4xy+e^x ,u(0,y),Ux(0,y)=1
Find the derivative of y=\sin 2x +3\cos 5x
Uxx +Uyy = Q(x,y)
U= C (constant)
this is possions equation ..u are requested to solve it by splitting the problem into two parts.
v----> non-homogeneous equation with homogeneous boundary conditions
w----> homogeneous equation with non homogeneous boundary condition
The solution of ∂2/∂x∂y (ϕA) at the point (2, -1, 1) of the function ϕ(x,y,z)=xy^2z and A=xzi-xy^2j+xz^2 is ________
A. 4i-2j
B. 5i+6j
C. 3i-2k
D. 3i+3j
Solve the given DE or IVP. First you need to determine what type of DE it is.

y'=sin^2 (3x-3y+1)
Solve d^2u/dx^2 + d^2u/dy^2 =0
Y = 0 , Y = 10 , X = infinity
u (x,y) = x - x ^2 at x = 10
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