A first order linear equation in the form y pxy fx can be

A first order linear equation in the form y\' + p(x)y = f(x) can be solved by finding an integrating factor mu(x) = exp(integral p(x)dx) Given the equation y\' + 5/x y = 10x^4 find mu(x) = exp(5ln(x)) Then find an explicit general solution with arbitrary constant C. y = Then solve the initial value problem with y(1) = 1 y =

Solution

1) exp(5 ln(x))=exp(ln(x^5))=x^5

2)

Multiplyng by this integrating factor gives

x^5y\'+5x^4y=10x^9

(yx^5)\'=10x^9

Integrating gives

yx^5=x^10+C

y=x^5+C/x^5

3)

y(1)=1=1+C

Hence, C=0

y=x^5