## The half-life of radioactive thorium-243 is 24 days. A sample involving 100 mg of thorium-243 is used in an experiment.

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(1) Solve the differential equation ( 2 x y − y x 2 + y 2 ) dx + ( x x 2 + y 2 − x 2 y 2 ) dy = 0.

(2) Solve the differential equation y ′ + y ln y = xy.

(3) Solve the differential equation (3 sin y − 5x) dx + 2x 2 cot ydy = 0.

(4) A block is released with an initial velocity v0 = 20 m/s from the bottom of an inclined plane making an angle of 30◦ with the horizontal. If the constant of friction is µ = 0.4, find the displacement of the block ignoring air resistance (g = 10 m/s2 ).

(5) The half-life of radioactive thorium-243 is 24 days. A sample involving 100 mg of thorium-243 is used in an experiment. If 1 mg of thorium-243 is added to the sample daily, find the amount of thorium-243 in the sample one month later.

(6) Solve the differential equation y ′′ − 7y ′ + 12y = e−x (17 − 42x) − e 3x .

(7) Solve the differential equation (D2 − 1) y = ex sec2 x tan x.

(8) Solve the differential equation y ′′ − 2y ′ + y = e x ln2 x x .

(9) A 10 kg weight is attached to the lower end of a spring suspended from the ceiling. The weight comes to rest in its equilibrium position, thereby stretching the spring 2 m. Then, beginning at t = 0, an external force given by F (t) = 10 cos 3t is applied to the system. The medium offers a resistance numerically equal to 4 times the instantaneous velocity of the weight. Assuming g = 10 m/s2 , find the displacement of the weight as a function of time.