Differential Equations Answers

a) Solve (𝐷

2 + 1)𝑦 = 3𝑥 − 8 cot 𝑥 where 𝐷 =

𝑑

𝑑𝑥

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

a) In a circuit containing inductance 𝐿, resistance 𝑅 and voltage 𝐸, the current 𝐼 is

given by 𝐸 = 𝑅 𝐼 + 𝐿

𝑑𝐼

𝑑𝑡

. Given 𝐿 = 640 ℎ , 𝑅 = 250 𝛺 and 𝐸 =

500 𝑣𝑜𝑙𝑡 . 𝐼 being 0 when 𝑡 = 0. Find the time 𝑡 that elapses, before the current

𝐼 reaches 90% of its maximum value.

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b) Solve the system:

𝑑𝑥

𝑑𝑡

+ 𝑥 − 𝑦 = 𝑡𝑒

𝑡

, 2𝑦 −

𝑑𝑥

𝑑𝑡

+

𝑑𝑦

𝑑𝑡

= 𝑒

Solve (𝐷

2 − 3𝐷 + 2)𝑦 = 𝑥

2 + sin 𝑥 where 𝐷 =

𝑑

𝑑𝑥

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 (D²-2DD')=x³y+e^5x

Shiw that the equations xp-yp=0, z(xp+yq)=2xy are compatible and solve them

A string of iength L is stretched and fastened to two fix points. Find the solution of

the r.{ave equatiorl (vibrating string) ytt = a^2.yxx, when initial displacernent

y(x,0) = f (x) = b sin (pi.x / t).

also find the Fourier cosine transformation of exp(-x^2)

solve the differential equation by the method of variation of parameters d²y/dx²+9y=sec3x