FVC I Exercise Sheet 2 (First order differential equations)
1. By separating the variables, find the general solution of each of the following equations.
dy
(a) xy 2 dx = 1 + y 3 , x > 0.
dy
(b) y cos2 x dx − tan x − 2 = 0, − 21 π < x < 12 π
dy
(c) dx
= y sin(2x + 3).
dy
(d) x sin y + (x2 + 1) cos y dx = 0, 0 < y < π2 .
2. Find the general solution of each of the following linear equations using the integrating
factor method.
dy
(a) x dx − 2y = x5 , x > 0.
dy
(b) x(x + 1) dx +y =x+1
dy 1
(c) dx
+ x+1
y = sin x
dy 4
(d) dx
+ x
y = x4
3. Solve each of the following equations.
dy
(a) x3 dx + (x2 + x)y = 1, y(1) = 0
dy
(b) x dx − 4y + 2x2 + 4 = 0, x 6= 0, y(1) = 1
4. Find the solution of the differential equation
dy 5xy
=
dx (x + 3)(2x + 1)
which satisfies the condition y = 3 when x = 0, expressing your answer in the form
y = φ(x).
5. Find the particular solution of the linear differential equation
dy
x cos x + (x sin x + cos x)y = sin x, 0 < x < π2 ,
dx
which satisfies the condition y = 0 when x = π/6.
6. Rewrite the following equations in the form y 0 = f (x, y) where f is a homogeneous
function of x and y. Then for each use an appropriate substitution to convert the
equation into a separable equation. You need not solve the equation.
dy
(a) (x2 + 2y 2 ) dx − xy = 0
dy
(b) x2 dx − 1 = y(x + y)
7. Convert the following Bernoulli equations into linear first-order equations. You need
not solve the equations.
dy
(a) 2 dx + y + (1 − x)y 3 = 0
dy
(b) dx
= e−x y 2 + ex y
1. By separating the variables, find the general solution of each of the following equations.
dy
(a) xy 2 dx = 1 + y 3 , x > 0.
dy
(b) y cos2 x dx − tan x − 2 = 0, − 21 π < x < 12 π
dy
(c) dx
= y sin(2x + 3).
dy
(d) x sin y + (x2 + 1) cos y dx = 0, 0 < y < π2 .
2. Find the general solution of each of the following linear equations using the integrating
factor method.
dy
(a) x dx − 2y = x5 , x > 0.
dy
(b) x(x + 1) dx +y =x+1
dy 1
(c) dx
+ x+1
y = sin x
dy 4
(d) dx
+ x
y = x4
3. Solve each of the following equations.
dy
(a) x3 dx + (x2 + x)y = 1, y(1) = 0
dy
(b) x dx − 4y + 2x2 + 4 = 0, x 6= 0, y(1) = 1
4. Find the solution of the differential equation
dy 5xy
=
dx (x + 3)(2x + 1)
which satisfies the condition y = 3 when x = 0, expressing your answer in the form
y = φ(x).
5. Find the particular solution of the linear differential equation
dy
x cos x + (x sin x + cos x)y = sin x, 0 < x < π2 ,
dx
which satisfies the condition y = 0 when x = π/6.
6. Rewrite the following equations in the form y 0 = f (x, y) where f is a homogeneous
function of x and y. Then for each use an appropriate substitution to convert the
equation into a separable equation. You need not solve the equation.
dy
(a) (x2 + 2y 2 ) dx − xy = 0
dy
(b) x2 dx − 1 = y(x + y)
7. Convert the following Bernoulli equations into linear first-order equations. You need
not solve the equations.
dy
(a) 2 dx + y + (1 − x)y 3 = 0
dy
(b) dx
= e−x y 2 + ex y
