Differential Equations Answers

yp-x²q²-x²y=0

1. Differentiate of the following functions with respect to x:

i) 𝑆𝑖𝑛𝑥∙ ln(5𝑥)

ii) √𝑐𝑜𝑠√𝑥

iii) 𝑒 ln (𝑡𝑎𝑛5𝑥)

iv) 𝑆𝑖𝑛2 {ln(𝑠𝑒𝑐𝑥)}

v) ln (𝑡𝑎𝑛𝑥) �

write the ordinary differential equation (1+sin y)dx = (2ycosy-x(secy-tan y))dy

A 2000 L tank initially contains 40 kg of salt dissolved in 1000 L of water. A brine solution containing 0.02 kg/L of salt flows into the tank at a rate of 50 L/min. The solution is kept thoroughly mixed, and the mixture flows out at a rate of 25 L/min. (a) Find the quantity of salt in the tank at any time t > 0) prior to overflow. (b) Find the time of overflow.

solve the differential equation to the indicated inintial conditions; "y+4y=0" "y(0)=4" "y(0)=6"

1. the differential equation(1+²"x" )"dy\/dx" +"xy" =0 , determine the particular solution when "y" (0)=2, express y interms of "x" .
2. show that "dy\/dx="e^(y-x) is seperable and find the general solution expessing y in terms of x
3. use the U substitution "y=tx" to solve "(x^6-2x^5+2x^4-y^3+4x^2y)dx+(xy^2-4x^3)dy=0"

Solve the following initial value problem

Ut(x,t)=10Uxx(x,t) -10

U(-1,t)=U(1,t) Ux(-1,t)=Ux(1,t) t>0

Ux(x,0)=x+1 -1

Let f(x) = (x^2−1)/(x^4+1)

(a) At which points does the graph of the f(x) have a horizontal tangent line?

(b) Draw the graph of f(x) on MATLAB(or octave online) and identify the points for horizontal tangents on the graph.

Determine the general solution to the equation ∂²u/∂t²=c²(∂²u/∂x²) under the boundary conditions u(0,t)=u(1,t)=0 and initial conditions u(x,0)=Φ(x), u_t(x,o)=Ψ(x)