Q.218. Differential equation.

Question 218.
Find the general solution of the differential equation (x + 2)^2*dy/dx = 5 - 8y - 4xy.

Answer 218.
(x + 2)^2 * dy/dx = 5 - 8y - 4xy
=> (x + 2)^2 dy/dx = 5 - 4y (x + 2)
=> dy/dx + y * 4/(x + 2) = 5/(x + 2)^2

This is a linear differential equation of first order of the form
dy/dx + Py = Q, where P and Q are functions of x alone.
To solve it, multiply both sides by
e^∫Pdx = e^∫4/(x+2) dx = e^4ln(x+2) = (x+2)^4

=> (x+2)^4 dy/dx + 4(x+2)^3 = 20(x+2)^2
=> d/dx [(x+2)^4 * y] = 20 (x+2)^2
=> d [(x+2)^4 y] = 20 (x+2)^2 dx

Integrating,
y * (x+2)^4 = (20/3) (x+2)^3 + c.

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Verification:
Differentiation the solution w.r.t. x,
(x+2)^4 dy/dx + 4(x+2)^3 * y = 20 (x+2)^2
=> (x+2)^2 dy/dx + 4y(x+2) = 5
=> (x+2)^2 dy/dx = 5 - 8y - 4xy
which is the given differential equation to be solved.
This proves that the general solution of the differential equation
obtained as above is correct.

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