Question 1

Q1: Determine order and degree (if defined) of the differential equation: $\frac{d^{4} y}{d x^{4}}+\sin \left(y^{\prime \prime \prime}\right)=0$

Accepted Answer

Given Differential Equation: $$\frac{d^{4} y}{d x^{4}}+\sin \left(y^{\prime \prime \prime}\right)=0$$ Order: The highest order derivative present in the equation is $\frac{d^{4} y}{d x^{4}}$. Therefore, the order of the differential equation is 4. Degree: The given differential equation is not a pol...

Question 2

Q10: Determine order and degree (if defined) of the differential equation: $y^{\prime \prime}+2 y^{\prime}+\sin y=0$

Accepted Answer

Given Differential Equation: $$y^{\prime \prime}+2 y^{\prime}+\sin y=0$$ Order: The highest order derivative present in the equation is $y''$. Therefore, the order of the differential equation is 2. Degree: The equation is a polynomial equation in its derivatives $y''$ and $y'$. The term $\sin y$ do...

Question 3

Q11: The degree of the differential equation $\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1...

Accepted Answer

Given Differential Equation: $$\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0$$ Degree: The degree of a differential equation is defined only if it is a polynomial equation in its derivatives. The given equation contains the term $\sin...

Question 4

Q12: The order of the differential equation $2 x^{2} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+y=0$ is (A) 2 (B) 1 (C) 0 (D) not defined

Accepted Answer

Given Differential Equation: $$2 x^{2} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+y=0$$ Order: The order of a differential equation is the order of the highest derivative present in the equation. The derivatives in the equation are $\frac{d^2y}{dx^2}$ (second order) and $\frac{dy}{dx}$ (first order)....

Question 5

Q2: Determine order and degree (if defined) of the differential equation: $y^{\prime}+5 y=0$

Accepted Answer

Given Differential Equation: $$y^{\prime}+5 y=0$$ This can be written as: $$\frac{dy}{dx} + 5y = 0$$ Order: The highest order derivative present in the equation is $y^{\prime}$ or $\frac{dy}{dx}$, which is of the first order. Therefore, the order of the differential equation is 1. Degree: The equati...

Question 6

Q3: Determine order and degree (if defined) of the differential equation: $\left(\frac{d s}{d t}\right)^{4}+3 s \frac{d^{2} s}{d t^{2}}=0$

Accepted Answer

Given Differential Equation: $$\left(\frac{d s}{d t}\right)^{4}+3 s \frac{d^{2} s}{d t^{2}}=0$$ Order: The derivatives present are $\frac{ds}{dt}$ (first order) and $\frac{d^2s}{dt^2}$ (second order). The highest order derivative is $\frac{d^{2} s}{d t^{2}}$. Therefore, the order of the differential...

Question 7

Q4: Determine order and degree (if defined) of the differential equation: $\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d x}\right)=0$

Accepted Answer

Given Differential Equation: $$\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d x}\right)=0$$ Order: The highest order derivative present in the equation is $\frac{d^{2} y}{d x^{2}}$. Therefore, the order of the differential equation is 2. Degree: The given differential equation is n...

Question 8

Q5: Determine order and degree (if defined) of the differential equation: $\frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x$

Accepted Answer

Given Differential Equation: $$\frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x$$ Order: The highest order derivative present in the equation is $\frac{d^{2} y}{d x^{2}}$. Therefore, the order of the differential equation is 2. Degree: The equation is a polynomial equation in its derivatives. The highest o...

Question 9

Q6: Determine order and degree (if defined) of the differential equation: $\left(y^{\prime \prime \prime}\right)^{2}+\left(y^{\prime \prime}\right)^{3}+\l...

Accepted Answer

Given Differential Equation: $$\left(y^{\prime \prime \prime}\right)^{2}+\left(y^{\prime \prime}\right)^{3}+\left(y^{\prime}\right)^{4}+y^{5}=0$$ Order: The derivatives present are $y'''$ (third order), $y''$ (second order), and $y'$ (first order). The highest order derivative is $y'''$. Therefore,...

Question 10

Q7: Determine order and degree (if defined) of the differential equation: $y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0$

Accepted Answer

Given Differential Equation: $$y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0$$ Order: The highest order derivative present in the equation is $y'''$. Therefore, the order of the differential equation is 3. Degree: The equation is a polynomial equation in its derivatives. The highest orde...

Question 11

Q8: Determine order and degree (if defined) of the differential equation: $y^{\prime}+y=e^{x}$

Accepted Answer

Given Differential Equation: $$y^{\prime}+y=e^{x}$$ Order: The highest order derivative present in the equation is $y'$. Therefore, the order of the differential equation is 1. Degree: The equation is a polynomial equation in its derivatives. The highest order derivative is $y'$, and its power is 1....

Question 12

Q9: Determine order and degree (if defined) of the differential equation: $y^{\prime \prime}+\left(y^{\prime}\right)^{2}+2 y=0$

Accepted Answer

Given Differential Equation: $$y^{\prime \prime}+\left(y^{\prime}\right)^{2}+2 y=0$$ Order: The highest order derivative present in the equation is $y''$. Therefore, the order of the differential equation is 2. Degree: The equation is a polynomial equation in its derivatives. The highest order deriv...

Question 13

Q1: Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y=e^{x}+1$ : $y^{\prime \prime}-y^{\p...

Accepted Answer

Given Function: $y = e^x + 1$ Given Differential Equation: $y'' - y' = 0$ To Verify: We need to show that the function satisfies the differential equation. Solution: First, find the derivatives of the given function. $$y = e^x + 1$$  Differentiating with respect to $x$: $$y' = \frac{d}{dx}(e^x + 1)...

Question 14

Q2: Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y=x^{2}+2 x+\mathrm{C}$ : $y^{\prime}...

Accepted Answer

Given Function: $y = x^2 + 2x + C$ Given Differential Equation: $y' - 2x - 2 = 0$ To Verify: We need to show that the function satisfies the differential equation. Solution: First, find the derivative of the given function. $$y = x^2 + 2x + C$$ Differentiating with respect to $x$: $$y' = \frac{d}{dx...

Question 15

Q3: Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y=\cos x+\mathrm{C}$ : $y^{\prime}+\s...

Accepted Answer

Given Function: $y = \cos x + C$ Given Differential Equation: $y' + \sin x = 0$ To Verify: We need to show that the function satisfies the differential equation. Solution: First, find the derivative of the given function. $$y = \cos x + C$$ Differentiating with respect to $x$: $$y' = \frac{d}{dx}(\c...

Question 16

Q4: Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y=\sqrt{1+x^{2}}$ : $y^{\prime}=\frac...

Accepted Answer

Given Function: $y = \sqrt{1+x^2}$ Given Differential Equation: $y' = \frac{xy}{1+x^2}$ To Verify: We need to show that the function satisfies the differential equation. Solution: First, find the derivative of the given function. $$y = (1+x^2)^{1/2}$$ Differentiating with respect to $x$: $$y' = \fra...

Question 17

Q5: Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y=\mathrm{A} x$ : $x y^{\prime}=y(x \...

Accepted Answer

Given Function: $y = Ax$ Given Differential Equation: $xy' = y$ (for $x 
eq 0$) To Verify: We need to show that the function satisfies the differential equation. Solution: First, find the derivative of the given function. $$y = Ax$$ Differentiating with respect to $x$: $$y' = \frac{d}{dx}(Ax) = A$$...

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