Q.333. Linear Differential Equation.

Question 333.
Solve the differential equation (2xy)dx + dy = (2x^3)dx.

Answer 333.
(2xy) dx + dy = (2x^3) dx
=> dy/dx + 2xy = 2x^3

This is a linear differential equation of the form
dy/dx + Py = Q, where P and Q are functions of x.
Such equations can be solved by multiplying the equation
with the integrating factor e^∫ Pdx = e^∫2x dx = e^(x^2)

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

Integrating,
y e^(x^2) = ∫ 2x^3 e^(x^2) dx ... (1)

Let x^2 = t
=> 2x dx = dt
=> ∫ 2x^3 e^(x^2) dx
= ∫ te^t dt
= t ∫e^t dt - ∫ [d/dt(t) ∫ e^t dt] dt ... [integrating by parts]
= te^t - e^t + c
= x^2 e^(x^2) - e^(x^2) + c

Plugging in (1),
y e^(x^2) = x^2 e^(x^2) - e^(x^2) + c
=> y = x^2 - 1 + c e^(-x^2).

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