Ordinary Differential Equation - T 1

Rsquared Mathematics

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Time: 45:00

Topic: Ordinary Differential Equation

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The Quiz contains a total of 10 Questions.
Part B(Single Correct) contains 5 questions, each of 3 marks and -0.75 marks for incorrect answers.
Part C(MSQ) contains 5 questions, each with 4.75 marks and no negative marking. The maximum score for the Test is 38.75. Best of Luck.

Part - B.

1. A differential equation is considered to be ordinary if it has

(a) One dependent variable
(b) More than one dependent variable
(c) One independent variable
(d) More than one independent variable

Explanation: A differential equation is considered to be ordinary if it has one independent variable. Ordinary differential equations can have as many dependent variables as needed. For example, the ordinary differential equations...

2. The solution of \(\frac{d y}{d x}=\frac{a x+b}{c y+d}\) represents a parabola if,

(a) \(a=1, b=2\)
(b) \(a=0, c\neq 0\)
(c) \(a=0, c=0\)
(d) \(a=1, c=1\)

Explanation: \((c y+d) \cdot d y=(a x+b) d x\) leads to \(a x^{2}-c y^{2}+2 b x-2 d y+g=0\). For this to be a parabola, either \(a\) or \(c\) should be zero.

3. Classify the following differential equation

\(e^{x} \frac{d y}{d x}+3 y=x^{2} y\)

(a) Separable and not linear
(b) Linear and not separable
(c) Both separable and linear
(d) Neither separable nor linear

Explanation: \(e^{x} \frac{d y}{d x}=y\left(x^{2}-3\right)\) is separable, and \(\frac{1}{y} \frac{d y}{d x}=\frac{\left(x^{2}-3\right)}{e^{x}}\) shows linearity. Therefore, the given differential equation is both separable and linear.

4. Solution of the differential equation \(\frac{d y}{d x}=e^{x}, y(\ln 3)=1\)

(a) is bounded on \(\mathbb{R}\)
(b) is not bounded on \(\mathbb{R}^{-}\)
(c) does not intersect \(\ln x-2
(d) Intersects \(||x|-1|, x \leq 0

Explanation: The solution to the differential equation is \(y=e^{x}-2\). The curve \(y=e^{x}-2\) does not intersect \(\ln x-2\), as \(e^{x}-2\) and \(||x|-1|\) clearly do not intersect in the given graphs.

[Graph] [Graph]

5. If \(f(x)=\log _{x}\{\ln (x)\}\) then \(f^{\prime}(x)\) at \(x=e\) is

(a) \(e\)
(b) \(-e\)
(c) \(e^{2}\)
(d) \(e^{-1}\)

Explanation: \(f(x)=\log _{x}\{\ln (x)\}=\frac{\ln (\ln (x))}{\ln (x)}\), and \(f^{\prime}(x)=\frac{1-\ln \{\ln (x)\}}{x\{\ln (x)\}^{2}}\). Evaluating at \(x=e\), \(f^{\prime}(e)=\frac{1}{e}\).

Part - C

6. The differential equation \(\left|\frac{d y}{d x}\right|+|y|+k=0\)

(a) has a finite set of solutions for \(k \geq 0\)
(b) has uncountable solutions for fixed \(k \in \mathbb{R}\)
(c) the set of functions which are not a solution of the given differential equation for some fixed \(k<0\) is uncountable.
(d) there does not exist any periodic function which is a solution of the given differential equation for some fixed \(k\).

Explanation: For \(k>0\), the differential equation has no solution, and for \(k=0\), the differential equation has a trivial solution. Only for some fixed \(k<0\), there are two solutions \(y= \pm k\).

7. Which of the following cannot be a solution of some differential equation on \(\mathbb{R}\).

(a) \([x]\)
(b) \(\frac{|x|}{x}\)
(c) \(f(x)= \begin{cases}x^{2} & \text { if } x<1 \\ x & \text { if } 1 \leq x<2 \\ 2 x-1 & \text { if } 2 \leq x\end{cases}\)
(d) \(\cos x+\sin \frac{1}{x}+e^{x}\)

Explanation: Solutions of a differential equation are continuous.

8. The solution of the differential equation \(\frac{d y}{d x}=x, y(0)=1\)

(a) is unbounded on \(\mathbb{R}\)
(b) is unbounded on \(\mathbb{R}^{+}\)
(c) \(\exists\) a point at which the value of the function is zero.
(d) is bounded below

Explanation: \(\frac{d y}{d x}=x\), \(y=\frac{x^{2}}{2}+c\), \(y(0)=1 \Rightarrow c=1\), \(y=\frac{x^{2}}{2}+1\). \(\lim _{x \rightarrow+\infty} y=+\infty\) and \(\lim _{x \rightarrow-\infty} y=+\infty\), so \(y\) is unbounded on \(\mathbb{R}\).

9. Let \(f(t)=\ln (t)\). Then \(\frac{d}{d x}\left(\int_{x^{2}}^{x^{3}} f(t) d t\right)\)

(a) has a value 0 when \(x=0\)
(b) has a value 0 when \(x=1\) and \(x=\frac{4}{9}\)
(c) has a value \(9 e^{2}-4 e\) when \(x=e\)
(d) has a differential coefficient \(27 e-8\) for \(x=e\)

Exp: Let \(F(x)=\frac{d}{d x}\left(\int_{x^{2}}^{x^{3}} \ln (t) d t\right)=3 x^{2} \cdot \log x^{3}-2 x \log x^{2}\).

\(F^{\prime}(x)=(9 x-4)+(18 x-4) \log x\)

Clearly, \(F(x)\) and \(F^{\prime}(x)\) satisfy (b),(c) and (d) respectively.

10. The function \(u(x)=e^x \sin(x); v(x)=e^x\cos(x)\) satisfy the equation -

(a) \(v \frac{d u}{d x}-u \frac{d u}{d x}=u^{2}+v^{2}\)
(b) \(\frac{d^{2} u}{d x^{2}}=2 v\)
(c) \(\frac{d^{2} v}{d x^{2}}=-2 u\)
(d) None of these

Exp: \(u=e^{x} \sin x \Rightarrow \frac{d u}{d x}=e^{x} \sin x+e^{x} \cos x=u+v\)

\(v=e^{x} \cos x \Rightarrow \frac{d v}{d x}=e^{x} \cos x-e^{x} \sin x=v-u\)

\(v \frac{d u}{d x}-u \frac{d v}{d x}=v(u+v)-u(v-u)=u^{2}+v^{2}\)

\(\frac{d^{2} u}{d x^{2}}=\frac{d u}{d x}+\frac{d v}{d x}=u+v+v-u=2 v\)

and \(\frac{d^{2} v}{d x^{2}}=-\frac{d u}{d x}+\frac{d v}{d x}=(v-u)-(v+u)=-2 u\)