For the differential equation $x\frac{dy}{dx}+3y= x^2\log_e x$, which of the following statements are TRUE?
(A) Product of order and degree is 1
(B) Integrating factor is $x^3$
(C) Integrating factor is $3x$
(D) General solution is $y =\frac{x^3}{36} (6\log_e|x|-1)+ Cx^{-3}$, C is an arbitrary constant.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
$\text{DE: }x\frac{dy}{dx}+3y=x^2\ln x\;\Rightarrow\;\frac{dy}{dx}+\frac{3}{x}y=x\ln x$
$\text{Order}=1,\;\text{Degree}=1\;\Rightarrow\;\text{product}=1$
$\text{IF}=e^{\int \frac{3}{x}dx}=x^3$
$x^3y=\int x^3\cdot(x\ln x)\,dx=\int x^4\ln x\,dx=\frac{x^5}{5}\ln x-\frac{x^5}{25}+C$
$\Rightarrow\;y=\frac{x^2}{5}\ln x-\frac{x^2}{25}+Cx^{-3}$
True statements: (A) and (B)