Linear differential equation

equation 4 – Linear differential
A first-order linear differential equation is an equation of the form
y^ +P(x)y=Q(x)
where P and Q are functions of x. An equation that is written in this form is said to be in standard form.
To solve a linear differential equation, write it in standard form to identify the
functions P(x) and Q(x). Then integrate P(x) and form the expression
u(x)=e^??? P(x)dx?
which is called an integrating factor. The general solution of the equation is
y=1/u(x) ???u(x) ? Q(x)dx
Example 1: Find the general solution of y^ +3y=e^2x
Solution
For this equation, P(x)=3 and Q(x)=e^2x.So,the integrating factor is
u(x)=e^??? P(x)dx?=e^(??3 dx)=e^3x (Integrating factor)
This implies that the general solution is
y=1/u(x) ???u(x) ? Q(x)dx
y=1/e^3x ???e^3x ? e^2x dx
y=1/e^3x ???e^5x ? dx
y=1/e^3x (( e^5x )/( 5 )+C)
y=( e^5x )/( 5 )+Ce^(-3x) (General solution)


Example 2: Find the general solution of xy^ -2y=x^2. Assume x>0
Solution; Begin by writing the equation in standard form
y^ +((-2 )/x)y=x Standard form, y^ +P(x)y=Q(x)
In this form, you can see that
P(x)=(-2 )/x and Q(x)=x .So,the integrating factor is
u(x)=e^??? P(x)dx?=e^(??(-2)?x dx)
=e^(-2 ln?x )=e^ln??x^(-2) ? =x^(-2) (Integrating factor)
This implies that the general solution is
y=1/u(x) ???u(x) ? Q(x)dx
y=1/x^(-2) ???x^(-2) ? x dx
y=x^2 ???1/x ? dx=x(ln?x+C)
y=x^2 ln?x+Cx^2 (General solution)




Example 3: Find the solution of the initial-value problem
x^2 y^ +xy=1 x>0 y(1)=2 .
Solution: We must first divide both sides by the coefficient of y^ to put the differential equation into standard form:
y^ +( 1 )/( x ) y=1/x^2
P(x)=( 1 )/x and Q(x)=1/x^2 .So,the integrating factor is
u(x)=e^??? P(x)dx?=e^(??1?x dx) =e^ln?x =x
This implies that the general solution is
y=1/u(x) ???u(x) ? Q(x)dx
y=1/( x ) ???x ? 1/x^2 dx
y=1/( x ) ?? 1/( x ) dx
y=1/( x ) (ln?x+c)
y(1)=2? 2=1(0+c)? c=2
y=1/( x ) (ln?x+2)




(1) Solve x y^ -3y=x^2
(2) Use an integrating factor to solve the first-order linear differential equation
x y^ +y=sin?x
(3) Solve y^ +y tan?x=sec?x with y(??4)=?2
(4) Solve x dy/dx+3y=sin?x/x^2 with y(0)=2