Match List-I with List-II
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List-I Differential Equation |
List-II Order and degree of differential equation |
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(A) $\frac{d^2y}{dx^2}+2(\frac{dy}{dx})^2=e^{\frac{dy}{dx}}+1$ |
(I) Order = 1, Degree = 2 |
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(B) $(\frac{d^2y}{dx^2})^2+4(\frac{dy}{dx})^3=e^y – 1$ |
(II) Order = 2, Degree = 1 |
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(C) $3 (\frac{dy}{dx})+4y+e^y=\frac{dy}{dx}$ |
(III) Order = 2, Degree = 2 |
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(D) $\frac{d^2y}{dx^2}+3(\frac{dy}{dx})= (e^y+\frac{dy}{dx})^2$ |
(IV) Order = 2, Degree = Not defined |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
Given differential equations and to find order and degree:
(A) \(\frac{d^2y}{dx^2} + 2 \left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1\)
- Order is highest derivative: \(\frac{d^2y}{dx^2}\) ⇒ Order = 2
- Degree is the power of highest order derivative after removing radicals and fractions of derivatives. Here, \(\frac{d^2y}{dx^2}\) is to the power 1 but RHS has exponential of \(\frac{dy}{dx}\), which is not a polynomial expression in derivatives ⇒ Degree is not defined.
(B) \(\left(\frac{d^2y}{dx^2}\right)^2 + 4 \left(\frac{dy}{dx}\right)^3 = e^y - 1\)
- Highest order derivative is \(\frac{d^2y}{dx^2}\) squared ⇒ Order = 2
- Degree is the power of highest order derivative after simplification ⇒ Degree = 2
(C) \(3 \frac{dy}{dx} + 4y + e^y = \frac{dy}{dx}\)
- Highest order derivative is \(\frac{dy}{dx}\) ⇒ Order = 1
- Degree is the power of \(\frac{dy}{dx}\) after simplification. Rearranging: \(3 \frac{dy}{dx} - \frac{dy}{dx} = -4y - e^y \Rightarrow 2 \frac{dy}{dx} = \ldots\), degree of \(\frac{dy}{dx}\) is 1, no powers, but the term \(e^y\) is not involving derivative.
But since \(\frac{dy}{dx}\) appears to power 1, degree = 1
(D) \(\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} = (e^y + \frac{dy}{dx})^2\)
- Highest order derivative is \(\frac{d^2y}{dx^2}\) ⇒ Order = 2
- Degree is power of \(\frac{d^2y}{dx^2}\) which is 1 (since it is not raised to any power)
- The RHS is a square of expression involving first derivative, but highest order derivative is second derivative raised to 1 only ⇒ Degree = 1
Matching:
- (A) → (IV) Order = 2, Degree = Not defined
- (B) → (III) Order = 2, Degree = 2
- (C) → (I) Order = 1, Degree = 1
- (D) → (II) Order = 2, Degree = 1