Question

The general solution of $x(x - 1)\frac{dy}{dx} = x^3(2x - 1) + (x - 2)y$ is

• A. $y(x - 1) = x^3 + c(x - 1)$, where c is the constant of integration.
• B. $y = x^3(x - 1) + c$, where c is the constant of integration.
• C. $y(x - 1) = x^3(x - 1) + cx^2$, where c is the constant of integration.
• D. $y(x - 1) = x^3(x - 1) + c$, where c is the constant of integration.

Hint:

Rewrite the equation as a linear differential equation in y and solve it using an integrating factor.

Answer 1

The Correct Option is C

Step 1: Verify the homogeneous part
For $y_c=\dfrac{x^2}{x-1}$, the homogeneous equation requires $\dfrac{y_c'}{y_c}=\dfrac{x-2}{x(x-1)}$.
The logarithmic derivative of $\dfrac{x^2}{x-1}$ is $\dfrac{2}{x}-\dfrac{1}{x-1}=\dfrac{x-2}{x(x-1)}$, which works.

Step 2: Particular part
The particular solution is $y=x^3$, obtained from $y(x-1)=x^3(x-1)$ with $c=0$.

Step 3: Conclusion
So $y=x^3+\dfrac{cx^2}{x-1}$, i.e. $y(x-1)=x^3(x-1)+cx^2$, the third option.

Final Answer:
The general solution is $y(x-1)=x^3(x-1)+cx^2$. \[ \boxed{\text{(C) }y(x-1)=x^3(x-1)+cx^2} \]