# Solve the following differential equation: dydxyxxdydx+yx=x3-3 - Mathematics and Statistics

Sum

Solve the following differential equation:

"dy"/"dx" + "y"/"x" = "x"^3 - 3

#### Solution

"dy"/"dx" + "y"/"x" = "x"^3 - 3   ...(1)

This is the linear differential equation of the form

"dy"/"dx" + "P" * "y" = "Q", where P = 1/"x" and Q = "x"^3 - 3

∴ I.F. = "e"^(int "Pdx") = "e"^(int 1/"x" "dx")

= "e"^(log "x") = x

∴ the solution of (1) is given by

y(I.F.) = ∫ Q. (I.F.)dx + c1

∴ "y" * "x" = int ("x"^3 - 3)"x" "dx" + "c"_1
∴ "xy" = int ("x"^4 - 3"x") "dx" + "c"_1

∴ "xy" = "x"^5/5 - 3 * "x"^2/2 + "c"_1

∴ "x"^2/5 - "3x"^2/2 - "xy" = "c", where c = - c1

∴ This is the general solution.

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
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