Differential Equations Answers

Solve the differential equation given by

(3x + 2y + y^2)dx + (x+4xy+5y^2)dy=0

. Solve the following homogeneous differential equation: (x + 2y)dx − xdy = 0

Solve the following homogeneous differential equation 2x³ y^' = y(2x² − y²) 

(4𝑒^−𝑦 𝑠𝑖𝑛 𝑥 − 1) 𝑑𝑥 − 𝑑𝑦 = 0

8, Consider a pendulum bob of length 10cm hung from a ceiling. What should we do to the length in order to vary the period by 2?

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.