Question

Consider the differential equation dy/dx=(4y)/(x+3).

Find the general solution in explicit form.
Find the particular solution in explicit form, if y(-2)=1.
Find the equation of the tangent line at (-2,1) to the particular solution.

Answer 1

a) y = c(x + 3)⁴

b) y = (x + 3)⁴

c) y = 4x + 9

Explanation:

dy/dx = 4y/(x+3)

1/y .dy = 4/(x+3) .dx

ln|y| = 4ln|x+3| + ln|c|

ln|y| = ln|x+3|⁴ + ln|c|

y = c(x+3)⁴

x = -2, y = 1

1 = c(-2+3)⁴

1 = c

y = (x+3)⁴

Gradient of the tangent:

dy/dx = 4(1)/(-2+3) = 4

y = 4x + c

1 = 4(-2) + c

c = 1 + 8

c = 9

y = 4x + 9