1Give the general solution of dydxy5x2 2 Find the specific s

Solution

Solve the separable equation ( dy(x))/( dx) = (x+2) (y(x)+5):

Divide both sides by y(x)+5:

(( dy(x))/( dx))/(y(x)+5) = x+2

Integrate both sides with respect to x:

integral (( dy(x))/( dx))/(y(x)+5) dx = integral (x+2) dx

Evaluate the integrals:

log(y(x)+5) = x^2/2+2 x+c_1, where c_1 is an arbitrary constant.

Solve for y(x):

y(x) = e^(x^2/2+2 x+c_1)-5

Simplify the arbitrary constant:

y(x) = c_1 e^(x^2/2+2 x)-5




Solve the separable equation ( dy(x))/( dx) = y(x)-2, such that y(0) = 6:

Divide both sides by y(x)-2:

(( dy(x))/( dx))/(y(x)-2) = 1

Integrate both sides with respect to x:

integral (( dy(x))/( dx))/(y(x)-2) dx = integral 1 dx

Evaluate the integrals:

log(y(x)-2) = x+c_1, where c_1 is an arbitrary constant.

Solve for y(x):

y(x) = e^(x+c_1)+2

Simplify the arbitrary constant:

y(x) = c_1 e^x+2

Solve for c_1 using the initial conditions:

Substitute y(0) = 6 into y(x) = c_1 e^x+2:

c_1+2 = 6

Solve the equation:

c_1 = 4

Substitute c_1 = 4 into y(x) = c_1 e^x+2:

y(x) = 4 e^x+2