Differential Equations Answers

solve the following differential equations

1. (1 + 2 𝑒 𝑥 𝑦) + 2𝑒 𝑥 𝑦 (1 − 𝑥 𝑦 ) 𝑑𝑦 𝑑𝑥 = 0 

2.1. 𝑡𝑎𝑛 𝑥 . 𝑠𝑖𝑛2 𝑦 𝑑𝑥 + 𝑐𝑜𝑠2 𝑥 . 𝑐𝑜𝑡 𝑦 𝑑𝑦 = 0

2. (1 + 2 𝑒 𝑥 𝑦) + 2𝑒 𝑥 𝑦 (1 − 𝑥 𝑦 ) 𝑑𝑦 𝑑𝑥 = 0

3. (𝑥 − 2𝑦 + 1) 𝑑𝑥 + (4𝑥 − 3𝑦 − 6) 𝑑𝑦 = 0

4. 𝑦(𝑥 3 𝑒 𝑥𝑦 − 𝑦) 𝑑𝑥 + 𝑥 (𝑦 + 𝑥 3𝑒 𝑥𝑦) 𝑑𝑦 = 0

5. 𝑦 2 𝑑𝑥 + (𝑥 2 − 𝑥𝑦 − 𝑦 2 ) 𝑑𝑦 = 0

6. (4𝑒 −𝑦 𝑠𝑖𝑛 𝑥 − 1) 𝑑𝑥 − 𝑑𝑦 = 0

7. 𝑥 4 𝑑𝑦 𝑑𝑥 + 𝑥 3𝑦 − 𝑠𝑒𝑐(𝑥𝑦) = 0

8. (1 + 𝑠𝑖𝑛 𝑦) 𝑑𝑥 𝑑𝑦 = 2𝑦 𝑐𝑜𝑠 𝑦 − 𝑥(𝑠𝑒𝑐 𝑦 + 𝑡𝑎𝑛 𝑦) 

The amount A in a fixed account at any time t is known to satisfy the the differential equation

d²A/dt²+tdA/dt+(t²-4)A=0. Express A as a series in powers of t.

Using Taylor series expression, find a power series solution for the equation

2x²d²y/dx²-xdy/dx+(x-5)y=0 in powers of (x-1) if y(1)=4 and y'(1)=2

The population P, t years after the initial observation is given by the formula:

p_t=(100000)/(2+3e^-0.05t)

Determine the exact size of the population and the time in years when the rate of growth is maximum

Find the relation of the variable w if the growth rate is given by

dw/dt=(kw(α-w))/α, k>0, at t=0,w=α/(1+β)

Using annihilator method obtain particular solution of y"'+y= cosx+sinx