Download MATHS PROBLEM SHEET 7 — FIRST ORDER ODE`S

MATHS PROBLEM SHEET 7 — FIRST ORDER ODE’S
1. In each case, find the differential equation whose general solution is
(i) y(x) = Ae2x + Be−x + 2x,
(ii) x(t) = Ae−t cos(3t + B),
where A and B are arbitrary constants.
2. Solve the following problems by separating the variables:
1
dy
= 1 + 2 with y(1) = 0,
(i)
dx
y
(ii)
dy
x2
=
dx
y(1 + x3 )
with y(0) = 1.
3. Find the general solution of the differential equations:
(i) x2
dy
+ xy − y 2 = 0,
dx
(ii)
dy
x2 + 3xy + y 2
=
.
dx
x2
dy
2x + 2y − 2
=
is not homogeneous in
dx
3x + y − 5
x and y, but becomes homogeneous in x and y upon the substitution x = x+x 0 ,
y = y + y0 provided that the constants x0 , y0 are suitably chosen. Hence find
the general solution of the equation.
4. Show that the differential equation
5. Solve the following linear differential equations :
dy
(i) (x2 + 1) + xy = x,
dx
dy
(ii) (x + a) − 3y = (x + a)5 ,
dx
dy
+ 2(tan x)y = sin x with y( π3 ) = 0.
(iii)
dx
1
dy
6. By substituting y = 1/u, show that the non-linear equation
+ 2xy = xy 2
dx
du
− 2xu = −x. Hence find the general solution
reduces to the linear equation
dx
of the non-linear equation.
7. Find the general solution of the following differential equations:
(i)
dy
y y2
= + 2,
dx
x x
(ii) xy
dy
x2 + 1
= 2
,
dx
y −1
(iii) x
dy
− y = y 2 e2x .
dx
[Hint for (iii): use the substitution y = 1/u.]
8. Find the solution of the following problems:
dy
(i) (1 + x2 ) + 4xy = 1 with y(0) = 1,
dx
x+y−1
dy
=
with y(2) = 1,
(ii)
dx
x+y+1
(iii)
y + e−x
dy
+y =
dx
x+1
with y(0) = 1.
ANSWERS
1 (i) y 00 − y 0 − 2y = −2 − 4x, (ii) x00 + 2x0 + 10x = 0.
2 (i) y − tan−1 y = x − 1, (ii) y 2 = 1 + 23 ln |1 + x3 | .
3 (i) y − 2x = kx2 y, (ii) x = (x + y)[k − ln |x|].
4 (y − x + 3)4 = k(y + 2x − 3).
5 (i) (y − 1)2 (x2 + 1) = k, (ii) 2y = (x + a)5 + k(x + a)3 ,
(iii) y = cos x(1 − 2 cos x).
2
6 y[1 + kex ] = 2.
7 (i) y ln |kx| = −x, (ii) y 4 − 2y 2 = 2x2 + 4 ln |x| + k, (iii) (k − e2x )y = 2x.
8 (i) (1 + x2 )2 y = x + 31 x3 + 1 , (ii) y + ln 31 (x + y) = x − 1,
(iii) y = e−x (2x + 1).
2