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EULER – CAUCHY EQUATIONS – MCQ – LEVEL – 2

EULER – CAUCHY EQUATIONS – MCQ – LEVEL – 2

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EULER – CAUCHY EQUATIONS – MCQ – LEVEL – 2 0%

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Question 1 of 10

1. Question
If $[(x + 1)^2 f(x) - g(x)]$ is the particular integral of the differential equation, $(x+1)^2 \dfrac{d^2 y}{dx^2}-3(x+1)\dfrac{dy}{dx}+4y=x^2$ then,
- $f(x) = \log(x + 1)^2 ~\&\ ~g (x) = 2x + 5$
- $f(x) = \log(x + 1) ~\&\ ~g(x) = 5x + 1$
- $f(x)=\dfrac{1}{2}[\log(x+1)]^2 ~\&\ ~g(x)=\dfrac{8x+7}{4}$
- $f(x)=\dfrac{\log (x+1)^2}{2} ~\&\ ~g(x)=\dfrac{2x+1}{3}$
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Question 2 of 10

2. Question
The particular integral of the differential equation $[x^2D^2+xD-1]y=x^2e^{2x}$ will be :
- $\dfrac{e^{^{2x}}}{8}(2-x^{-1})$
- $\dfrac{e^{^{2x}}}{4}(2-x^{-1})$
- $\dfrac{e^{^{2x}}}{2}(2-x^{-1})$
- None of these
Correct

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Question 3 of 10

3. Question
The particular integral of the differential euqation, $x^3\dfrac{d^3y}{dx^3}+2x\dfrac{dy}{dx}-2y=x^2\log x + 3x$ is :
- $\dfrac{x^2}{2}(\log x-1)+ 2 \log x$
- $\dfrac{x^2}{2}(\log x-2)+ 3x \log x$
- $\dfrac{x^2}{2}(\log x-3)+3x \log x$
- $x(\log x-3)+ \dfrac{\log x}{x}$
Correct

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Question 4 of 10

4. Question
What would be the solution be the differential equation, $[x^2D^2-(2m-1)xD+(m^2+n^2)]y=n^2x^m\log x$ ?
- $y = [C_1\cos (n \log x) + C_2 \sin (n \log x)] x^m + x^m \log x$
- $y = [C_1 \cos (m \log x) + C_2 \sin (m \log x)] x^n + x^n \log x$
- $y = [C_1 \cos (\log x) + C_2 \sin (\log x)]x^m + x^n \log x$
- None of these
Correct

Incorrect
Question 5 of 10

5. Question
The general solution of the differential equation, $\dfrac{d^3y}{dx^3}-\dfrac{4}{x}\dfrac{d^2 y}{dx^2}+\dfrac{5}{x^2}\dfrac{dy}{dx}-\dfrac{2y}{x^3}=1$ will be given by :
- $y=C_1x^2+C_2x^{(5+\sqrt{21})/2}+C_3x^{(5-\sqrt{21})/2}-\dfrac{x^3}{5}$
- $y=C_1e^{2x}+C_2x^{(1+\sqrt{2})/2}+C_3x^{(1-\sqrt{2})/2}+\dfrac{x^3}{2}$
- $y=C_1x^3+C_2x^{(\sqrt{3}+1)/2}+C_3x^{(\sqrt{3}-1)/2}-\dfrac{x^3}{3}$
- None of these
Correct

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Question 6 of 10

6. Question
The general solution of the differential equation, $x^2\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+2=x^{-1}$ will be given by :
- $y = [C_1 \cos (\log x) + C_2 \sin (\log x)]$
- $y = [C_1 \cos \log x + C_2 \sin \log x) e^x + x^{-1}$
- $y = [C_1 \cos (\log x) + C_2 \sin (\log x)] x + x^{-2}$
- $y = [C_1 \cos (\log x) + C_2 \sin (\log x)]x + (5x)^{-1}$
Correct

Incorrect
Question 7 of 10

7. Question
What substitution will reduce the differential of the form, $(ax+b)^n\dfrac{d^n y}{dx^n}+a_1(ax+b)^{n-1}\dfrac{d^{n-1}y}{dx^{n-1}}+...+a_{n-1}(ax+b)\dfrac{dy}{dx}+(ax+b)y=X$
- $ax + b = \log z$
- $z = \log (ax + b)$
- $ax = \log z$
- $z = \log ax$
Correct

Incorrect
Question 8 of 10

8. Question
Solving $(x+a)^2\dfrac{d^2y}{dx^2}-4(x+a)\dfrac{dy}{dx}+6y=x,$ will give :
- $C_1(x+a)^2+C_2(x+a)^4+\dfrac{3x+5a}{6}$
- $C_1(x+a)+C_2(x+a)^2+\dfrac{2x+7a}{4}$
- $C_1(x+a)^2+C_2\log(x+a)+\dfrac{x+3a}{4}$
- $C_1(x+a)^2+C_2(x+a)^3+\dfrac{3x+2a}{6}$
Correct

Incorrect
Question 9 of 10

9. Question
The particular integral of the differential equation, $(x^2D^2 + 4xD + 2) y = e^x$ will be :
- $\dfrac{e^x}{x}$
- $xe^x$
- $\dfrac{e^x}{x^2}$
- $x^2e^x$
Correct

Incorrect
Question 10 of 10

10. Question
The PI of the differential equation, $[(1 + 2x)^2D^2 - 6(1 + 2x) D + 16]y = 8(1+2x)^2$ will be :
- $(1 + 2x) \log (1 + 2x)$
- $(1+2x)^2 \log (1 + 2x)$
- $e^{2x}[\log (1 + 2x)]^2$
- $(1 + 2x)^2 [\log (1 + 2x)]^2$
Correct

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