MTH401 Solution Spring 2026

MTH401 Assignment 01 Spring 2026 · Deadline: 12 May,2026 · Total marks= 20 · Name: · Student id:

Find the orthogonal trajectories of the family of parabolas

cx

where 𝑐is an arbitrary constant.

this is deferential equation

replace

with

now apply integration method

now multiply by 2 2y dy cx dx c

dy dx

dy dx

dy dx

than dy dx now y d xdx

ydy xdx

c

c c hence c according to given statement the logistic equation is

is constant

kt dx kdt

ln x ln kt c

ln kt c

ce dx kx dt k

dx kdt

now we use x(3)=40 condition

40(1 kt kt kt kt kt kt kt

kt

kt

k k x ce xce ce ce ce ce ce

ae

a a a x t e

e e

now for x(7)

1. 000164 799 0.000164 0.131
2. 131
3. 131 k

k k

k k k

k e e e

e e e

e

1. 131 approximately

Find the general solution of the differential equation. Identify 𝑦1, 𝑦2 and then show that both are linearly independent by Wronskian.

suppose that

now for m

real roots

now identify linear independence by using wronskian w(y ,y ) mx mx mx mx

m x m x e m e m e e m

m m here m m

c e c e c e c e e e

w

and y are linearly independent y=c c y y calculate e e e e w e e e e w e e

w w

e e

Find the second solution of the following differential equation by using the formula.

0, x y xy