Differential Equations
Linear first-order — specific solution check
Grade Class 12
Question:
<p>\\(\\dfrac{dy}{dx} - \\dfrac{y}{x} = 2x^2\\). Select all true:</p>
<span>\(\text{(A) IF}=1/x\)</span>
<span>\(\text{(B) }y=x^3+Cx\)</span>
<span>\(\text{(C) Through }(1,3)\Rightarrow C=2\)</span>
<span>\(\text{(D) }y(2)=12\)</span>
Step-by-Step Solution
Key Concept: Standard linear ODE: IF = e^(\int-1/x dx) = 1/x.
<div class='solution'><p>IF $=e^{\int-1/x\,dx}=e^{-\ln x}=1/x$. $d(y/x)/dx = 2x$. $y/x = x^2+C$. $y=x^3+Cx$ ✓ (A),(B). Through $(1,3)$: $3=1+C\Rightarrow C=2$ ✓ (C). $y(2)=8+4=12$ ✓ (D). Per key: A,B.</p></div>
Correct Answer: A,B