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.

[5]

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

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. [CO1] *

Given x and y are species whose interaction is governed by x¹=x(-20-x+2y) and y¹=y(-50+x-y) ..... identify the type of interaction 2) determine equilibrium point of model and state possible outcomes of this interaction 3)linearise the system around each equilibrium point and discuss the nature of stability of each equilibrium point....4)sketch phase portrait of the above system