Level 1 — RecognitionOrdinary Differential Equations

Ordinary Differential Equations

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Level: 1 — Recognition (MCQ + Matching + True/False with justification) Time Limit: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each) [10 marks]

Choose the single best answer.

Q1. The order and degree of the ODE $\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^5 + y = 0$ are: (a) 3 and 2 (b) 2 and 3 (c) 2 and 5 (d) 3 and 5

Q2. Which of the following equations is linear? (a) $y' + y^2 = x$ (b) $yy' = x$ (c) $y' + xy = \sin x$ (d) $y' + \cos y = 0$

Q3. The integrating factor of $\dfrac{dy}{dx} + 2xy = x$ is: (a) $e^{x^2}$ (b) $e^{2x}$ (c) $x^2$ (d) $e^{x^2/2}$

Q4. The characteristic equation of $y'' - 5y' + 6y = 0$ has roots: (a) $2, 3$ (b) $-2, -3$ (c) $1, 6$ (d) $-1, -6$

Q5. The general solution of $y'' + 4y = 0$ is: (a) $c_1 e^{2x} + c_2 e^{-2x}$ (b) $c_1\cos 2x + c_2\sin 2x$ (c) $(c_1 + c_2 x)e^{2x}$ (d) $c_1 e^{2x}\cos x$

Q6. The Laplace transform $\mathcal{L}\{e^{at}\}$ (for $s>a$ ) equals: (a) $\dfrac{1}{s+a}$ (b) $\dfrac{1}{s-a}$ (c) $\dfrac{a}{s^2+a^2}$ (d) $\dfrac{s}{s^2-a^2}$

Q7. The equation $M\,dx + N\,dy = 0$ is exact if and only if: (a) $M_x = N_y$ (b) $M_y = N_x$ (c) $M_y = -N_x$ (d) $MN = 1$

Q8. A Bernoulli equation $y' + P(x)y = Q(x)y^n$ is reduced to linear form by the substitution: (a) $v = y^n$ (b) $v = y^{1-n}$ (c) $v = e^y$ (d) $v = \ln y$

Q9. For the linear system $\mathbf{x}' = A\mathbf{x}$ , if the eigenvalues of $A$ are complex with positive real part, the critical point at the origin is a: (a) stable spiral (b) unstable spiral (c) centre (d) saddle

Q10. The Cauchy–Euler equation $x^2 y'' + a x y' + b y = 0$ is solved by trying: (a) $y = e^{mx}$ (b) $y = x^m$ (c) $y = \sin mx$ (d) $y = m^x$

Section B — Matching (1 mark each) [8 marks]

Match each item in Column X with the correct entry in Column Y.

#	Column X		Column Y
Q11	$\mathcal{L}\{\sin at\}$	P	$\dfrac{s}{s^2+a^2}$
Q12	$\mathcal{L}\{\cos at\}$	Q	$\dfrac{n!}{s^{n+1}}$
Q13	$\mathcal{L}\{t^n\}$	R	$\dfrac{a}{s^2+a^2}$
Q14	$\mathcal{L}\{1\}$	S	$\dfrac{1}{s}$

#	Column X (equation)		Column Y (name/type)
Q15	$(1-x^2)y'' - 2xy' + n(n+1)y = 0$	P	Bessel's equation
Q16	$x^2y'' + xy' + (x^2-\nu^2)y = 0$	Q	Separable equation
Q17	$\dfrac{dy}{dx} = g(x)h(y)$	R	Legendre's equation
Q18	$\mathcal{L}\{f * g\}$	S	$F(s)G(s)$

Section C — True/False WITH Justification (2 marks each: 1 verdict + 1 reason) [12 marks]

Q19. An autonomous ODE is one in which the independent variable does not appear explicitly. (T/F + justify)

Q20. The Picard–Lindelöf theorem guarantees a unique solution provided $f(x,y)$ is merely continuous. (T/F + justify)

Q21. If $y_1$ and $y_2$ are solutions of a homogeneous linear ODE, then $c_1y_1 + c_2y_2$ is also a solution. (T/F + justify)

Q22. The second shift (Heaviside) theorem states $\mathcal{L}\{u(t-a)f(t-a)\} = e^{-as}F(s)$ . (T/F + justify)

Q23. The Dirac delta function $\delta(t)$ satisfies $\int_{-\infty}^{\infty}\delta(t)\,dt = 0$ . (T/F + justify)

Q24. For a repeated characteristic root $m$ , the second independent solution is $x e^{mx}$ . (T/F + justify)

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (b) 2 and 3. Highest derivative is $d^2y/dx^2$ → order 2. Its highest power (after clearing radicals; none here) is 3 → degree 3.

Q2 — (c) $y' + xy = \sin x$ . Linear means $y$ and its derivatives appear to first power, no products/nonlinear functions. (a) has $y^2$ , (b) has $yy'$ , (d) has $\cos y$ — all nonlinear.

Q3 — (a) $e^{x^2}$ . IF $= e^{\int 2x\,dx} = e^{x^2}$ .

Q4 — (a) $2,3$ . $m^2 - 5m + 6 = (m-2)(m-3)=0$ .

Q5 — (b) $c_1\cos 2x + c_2\sin 2x$ . Roots $m=\pm 2i$ ⇒ oscillatory solution with $\omega=2$ .

Q6 — (b) $\dfrac{1}{s-a}$ . $\int_0^\infty e^{-(s-a)t}dt = 1/(s-a)$ , $s>a$ .

Q7 — (b) $M_y = N_x$ . Exactness condition (equality of mixed partials of the potential).

Q8 — (b) $v = y^{1-n}$ . Standard Bernoulli substitution linearizes the equation.

Q9 — (b) unstable spiral. Complex eigenvalues ⇒ spiral; positive real part ⇒ trajectories spiral outward ⇒ unstable.

Q10 — (b) $y = x^m$ . Equidimensional form; substituting gives an indicial (auxiliary) equation in $m$ .

Section B (1 mark each)

Q11 → R $\;\dfrac{a}{s^2+a^2}$
Q12 → P $\;\dfrac{s}{s^2+a^2}$
Q13 → Q $\;\dfrac{n!}{s^{n+1}}$
Q14 → S $\;\dfrac{1}{s}$
Q15 → R Legendre's equation
Q16 → P Bessel's equation
Q17 → Q Separable equation
Q18 → S $F(s)G(s)$ (convolution theorem)

Section C (2 marks each: 1 verdict + 1 justification)

Q19 — TRUE. Autonomous: $y' = f(y)$ ; the independent variable $x$ (or $t$ ) does not appear explicitly on the RHS. (+1 verdict, +1 reason)

Q20 — FALSE. Continuity of $f$ gives existence (Peano) but uniqueness requires $f$ to also be Lipschitz in $y$ (or $\partial f/\partial y$ continuous). (+1/+1)

Q21 — TRUE. By the superposition principle, linear combinations of solutions to a homogeneous linear ODE are solutions (linearity of the operator $L$ : $L[c_1y_1+c_2y_2]=c_1L[y_1]+c_2L[y_2]=0$ ). (+1/+1)

Q22 — TRUE. Second shift theorem: $\mathcal{L}\{u(t-a)f(t-a)\}=e^{-as}F(s)$ for $a\ge0$ , derived by substitution $\tau=t-a$ in the transform integral. (+1/+1)

Q23 — FALSE. $\int_{-\infty}^{\infty}\delta(t)\,dt = 1$ (unit impulse), not $0$ . (+1/+1)

Q24 — TRUE. For repeated root $m$ , reduction of order yields the second solution $xe^{mx}$ ; general solution $(c_1+c_2x)e^{mx}$ . (+1/+1)

[
  {"claim":"Q4 roots of m^2-5m+6 are 2 and 3","code":"m=symbols('m'); sol=solve(m**2-5*m+6,m); result=(set(sol)=={2,3})"},
  {"claim":"Q3 integrating factor exp(integral 2x)=exp(x^2)","code":"x=symbols('x'); IF=exp(integrate(2*x,x)); result=(simplify(IF-exp(x**2))==0)"},
  {"claim":"Q6 Laplace of e^{at} is 1/(s-a)","code":"t,s,a=symbols('t s a',positive=True); F=integrate(exp(a*t)*exp(-s*t),(t,0,oo)); result=(simplify(F-1/(s-a))==0)"},
  {"claim":"Q5 y=c1 cos2x+c2 sin2x solves y''+4y=0","code":"x,c1,c2=symbols('x c1 c2'); y=c1*cos(2*x)+c2*sin(2*x); result=(simplify(diff(y,x,2)+4*y)==0)"}
]