MATH 225: DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA ...

2.b) What conditions must b1,b2, and b3 satisfy in order for the system
x1 + x2 + 2x3 = b1,
x1
to be consistent. Find the solutions.
+ x3 = b2,
2x1 + x2 + 3x3 = b3
Solution :
⎡
1
⎣1
1
0
2
1
⎤
b1
b2⎦
⎡
1
→ ⎣0
1
−1
2
−1
⎤
b1
b2 − b1⎦
⎡
1
→ ⎣0
1
−1
2
−1
⎤
b1
b2 − b1 ⎦
2 1 3 b3 2 1 3 b3 0 −1 −1 b3 − 2b1
⎡
1
→ ⎣0
1
1
2
1
b1
b1 − b2
⎤
⎦
0 0 0 b1 − b3 + b2
Hence for consistency b3 = b1 + b2. In this case there are infinitely many solutions
x1 = b2 − t, x2 = b1 − b2 + t, x3 = t where t ∈ R is an arbitrary real variable.
3.a) Solve the differential equation x 2 y ′ + 2xy = 5y 4 with y(1) = 1. Discus the
existence and uniqueness of the solution of the differential equation with the initial
condition y(0) = 0.
Solution : Let v(x) = y −3 then
v ′ − 6
x
15
v + = 0
x2 This is a linear equation with the integrating factor ρ(x) = 1
x 6 . Hence the solution
is
v(x) = C x 6 + 15
7
1
x
�
5x
→ y(x) =
7Cx7 + 15
where C is an arbitrary constant. If y(1) = 1 then C = − 8
7 .
The singular solution of this equation is y = 0. Furthermore at x = 0 the existence
and uniqueness is not guaranteed. It is clear from the solution that y(0) = 0 is
satisfied for all C, hence there are infinitely many solutions.
3.b) Solve the differential equation (x + ln y)dx + ( x
y + ey )dy = 0 with y(0) = 1
Solution: M(x, y) = x + ln y and N(x, y) = x
y + ey . Then
but
Fx = M(x, y) = x + ln y → F (x, y) = x2
2
� 1/3
+ x ln y + g(y),
Fy = x
y + g′ = x
y + ey , → F (x, y) = x2
+ x ln y + ey
2