AP EAMCET · Maths · Differential Equations

Observe the following statements
I. If \(d y+2 x y d x=2 e^{-x^2} d x\), then \(y e^{x^2}=2 x+c\)
II. If \(y e^{-x^2}-2 x=c\), then \(d x=\left(2 e^{-x^2}-2 x y\right) d y\) which of the following is a correct statement?

A Both I and II are true
B Neither I nor II is true
C I is false, but II is true
D I is false, II is true

Verified Solution

Answer & Solution

Correct Answer

(C) I is false, but II is true

Step-by-step Solution

Detailed explanation

I. \(d y+2 x y d x=2 e^{-x^2} d x\) \(\Rightarrow \quad \frac{d y}{d x}+2 x y=2 e^{-x^2}\) This is a linear differential equation in \(y\) Here, \(P=2 x, Q=2 e^{-x^2}\) \(\therefore \text { I.F. }=e^{\int P d x}=e^{\int 2 x d x}=e^{x^2}\) \(\therefore\) Complete solution is…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Observe the following statementsI. If \(d y+2 x y d x=2 e^{-x^2} d x\), then \(y e^{x^2}=2 x+c\)II. If \(y e^{-x^2}-2 x=c\), then \(d x=\left(2 e^{-x^2}-2 x y\right) d y\) which of the following is a correct statement?

Step-by-step Solution

Observe the following statements
I. If \(d y+2 x y d x=2 e^{-x^2} d x\), then \(y e^{x^2}=2 x+c\)
II. If \(y e^{-x^2}-2 x=c\), then \(d x=\left(2 e^{-x^2}-2 x y\right) d y\) which of the following is a correct statement?