Differential Equations Answers

I.In each of Problems 23 through 30, use the method of reduction of order to find a second solution of the given differential equation.

1.  t2y″ − 4ty′ + 6y = 0, y1(t) = t2

2.  xy″ − y′ + 4x3y = 0, x > 0; y1(x) = sin x2

A spring is such that a 16 lb weight stretches it by 1.5 in. The weight is pulled down to a point

4 in below the equilibrium point and given an initial downward velocity of 4 ft/sec. An impressed

force of F(t) = 2 cos 74t is acting on the spring. Describe the motion.

Find the integral surface of the linear partial differential equation x(x^2+z)p - y(y^2+z)q = (x^2-y^2)z; p, q has their usual meaning , which contains the straight line

Solve (𝐷

2 − 3𝐷 + 2)𝑦 = 𝑥

2 + sin 𝑥 where 𝐷 =

𝑑

𝑑𝑥

a) A thin metal plate bounded by the 𝑥-axis and the lines 𝑥 = 0 and 𝑥 = 1 and

stretching to infinity in the 𝑦-direction has its upper and lower faces perfectly

insulated and its vertical edges and edge at infinity are maintained at the constant

temperature 0 °𝐶, while over the base temperature of 50 °𝐶 is maintained. Find

the steady state temperature 𝑢(𝑥,𝑡).

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b) If 𝑓(𝑥𝑦^2, 𝑧 − 2𝑥) = 0 then prove that 𝑥 𝑧x −1/2𝑦 𝑧𝑦 = 2 𝑥

Solve (𝐷^2 − 3𝐷 + 2)𝑦 = 𝑥^2 + sin 𝑥 where 𝐷 =𝑑/𝑑𝑥

Given x'=x(-20-x+2y)

y’=y(-50+x-y) identify the type of interaction represented by the system

solve the following differential equation: 2x^2y(d^2y/dx^2)+4y^2=x^2(dy/dx)^2+2xy(dy/dx)

Find orthogonal trajectory to the curve given by 𝑟 = 𝑎(1 + cos 𝜃)

Solve (𝐷

2 − 3𝐷 + 2)𝑦 = 𝑥

2 + sin 𝑥