Q.98. Differential Equation.

Question 98.
Solve xdy/dx - 3y = x² using the method of substitution.

Answer 98.
Let y = tx^3
=> dy/dx = x^3*dt/dx + 3x^2*t

Plugging y and dy/dx in the given equation x dy/dx - 3y = x^2,
=> x [x^3*dt/dx + 3x^2*t] - 3 (tx^3) = x^2
=> x^4 dt/dx + 3tx^3 - 3tx^3 = x^2
=> dt/dx = 1/x^2
 Integrating,
t = - 1/x + c
=> y/x^3 = - 1/x + c
=> y = cx^3 - x^2.

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