This quiz works best with JavaScript enabled. Home > Class 12 > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 19 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 19 (25 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Superposition principle applies to? A) Nonlinear systems. B) Linear homogeneous systems. C) Nonhomogeneous nonlinear systems. D) All differential equations. Show Answer Correct Answer: B) Linear homogeneous systems. 2. What does the natural log function become when exponentiated? A) A new constant. B) The variable y. C) The constant e. D) The original variable before taking the natural log. Show Answer Correct Answer: D) The original variable before taking the natural log. 3. Separate the variables for the equation $(x-1)\frac{dy}{dx}=(y+1)$ A) $\frac{1}{(x-1)}dy=\frac{1}{(y+1)}dx$. B) $\frac{1}{(y+1)}dy=\frac{1}{(x-1)}dx$. C) $(y+1)dy=(x-1)dx$. D) $\frac{x}{(y+1)}dy=dx$. Show Answer Correct Answer: B) $\frac{1}{(y+1)}dy=\frac{1}{(x-1)}dx$. 4. The Wronskian of two functions $y_1, y_2$ A) $y_1y_2$. B) $y_1'y_2-y_1y_2'$. C) $y_1+y_2$. D) None. Show Answer Correct Answer: B) $y_1'y_2-y_1y_2'$. 5. The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Find the general solution. A) P=.05e$^{1000t}$. B) P=1000e$^{5t}$. C) P=1000e$^{.05t}$. D) P=lne$^{1000t}$. Show Answer Correct Answer: C) P=1000e$^{.05t}$. 6. The Fourier cosine Transform of $e^{-ax}$ A) $F_s\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{a}{a^2+s^2}$. B) $F_s\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{s}{a^2+s^2}$. C) $F_c\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{a}{a^2+s^2}$. D) None of the above. Show Answer Correct Answer: C) $F_c\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{a}{a^2+s^2}$. 7. Which one of the following differential equations is linear? A) $x\\frac{d^2y}{dx^2}+\frac{dy}{dx}+3y=e^{2x}$. B) $\frac{d^2y}{dx}-\frac{dy}{dx}+\sin y=0$. C) $3\\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2-y=\cos x$. D) $2y\\frac{d^2y}{dx^2}+4\\frac{dy}{dx}+xy=5$. Show Answer Correct Answer: A) $x\\frac{d^2y}{dx^2}+\frac{dy}{dx}+3y=e^{2x}$. 8. The matrix exponential method is primarily used to ..... A) Solve nonlinear systems of equations. B) Solve systems of linear differential equations. C) Find eigenvalues of matrix. D) Compute determinants of matrices. Show Answer Correct Answer: B) Solve systems of linear differential equations. 9. The particular integral of the equation $\left(D^3-1\right)y=\left(e^x+1\right)^2$ A) $\frac{e^{-2x}}{7}+\frac{2x}{3}-1$. B) $\frac{e^{3x}}{7}+\frac{2x}{3}-1$. C) $\frac{e^{2x}}{7}+\frac{5x}{3}-1$. D) $\frac{e^{2x}}{7}+\frac{2x}{3}-1$. Show Answer Correct Answer: D) $\frac{e^{2x}}{7}+\frac{2x}{3}-1$. 10. Each ratio in equation dx/P=dy/Q=dz/R is equal to A) Dx+dy+dz/P+Q+R. B) Ldx+mdy+ndz/lP+mQ+nR. C) Pdx+Qdy+Rdz/p$^{2}$+Q$^{2}$+R$^{2}$. D) Dx/P=dy/Q=dz/R. Show Answer Correct Answer: B) Ldx+mdy+ndz/lP+mQ+nR. 11. A population of bacteria grows according to the differential equation dp/dt = 0.05p, where p(0) = 100. What is the population after 10 hours? A) 75.10. B) 200.35. C) 164.87. D) 50.25. Show Answer Correct Answer: C) 164.87. 12. If $\frac{dy}{dx}=4x-1$ $y\left(1\right)=3$ $y$ A) $2x^2-x$. B) $2x^2-x+2$. C) $2x^2-x+1$. D) $2x^2+1$. Show Answer Correct Answer: B) $2x^2-x+2$. 13. What is a homogeneous PDE? A) A homogeneous PDE has no free terms and all terms involve the dependent variable and its derivatives. B) A homogeneous PDE only involves constants and no variables. C) A homogeneous PDE has varying coefficients and free terms. D) A homogeneous PDE is defined by its non-linear terms. Show Answer Correct Answer: A) A homogeneous PDE has no free terms and all terms involve the dependent variable and its derivatives. 14. If $y=\frac{1}{e^{2x}}\left[\frac{x^4}{4}+\ln\left|x\right|+c\right]$ $c$ $x=1, \\y=0$ A) 0. B) $-\frac{1}{4}$. C) $\ln4$. D) Any value in Real Number. Show Answer Correct Answer: B) $-\frac{1}{4}$. 15. Which of the following is not Dirichlet's condition for the Fourier series expansion? A) F(x) is periodic, single valued, finite. B) F(x) has a finite number of discontinuities in only one period. C) F(x) has a finite number of maxima and minima. D) F(x) is a periodic, single valued, infinite. Show Answer Correct Answer: D) F(x) is a periodic, single valued, infinite. 16. When the dependent variable and all of its derivatives are of the first degree, the equation is A) Non linear. B) Linear. C) Both linear and non linear. D) None of the options. Show Answer Correct Answer: B) Linear. 17. An equivalent representation of the definite integral $\int_1^32x\cos\left(x^2\right)dx$ A) $\int_1^3\cos udu$. B) $\int_1^9\cos udu$. C) $\int_1^{\sqrt[]{3}}\cos udu$. D) $\int_1^92\sqrt[]{u}\cos udu$. Show Answer Correct Answer: B) $\int_1^9\cos udu$. 18. Solve the initial value problem $y'=x^3\left(1-y\right)\\\\, \\\\\\y\left(0\right)=3$ A) $y=e^{-x^4}+8$. B) $y=2e^{\frac{-x^4}{4}}+1$. C) $y=\frac{-x^4}{4}+C$. D) $y=e\frac{-x^4}{2}+1$. Show Answer Correct Answer: B) $y=2e^{\frac{-x^4}{4}}+1$. 19. Apply the initial condition $y(0) = 2$ $y = Ce^{3x}$ $\frac{dy}{dx} = 3y$ $C$ A) $C = 0$. B) $C = 1$. C) $C = 2$. D) $C = 3$. Show Answer Correct Answer: C) $C = 2$. 20. The general solution of the DE $\frac{d^2y}{dx^{2}}+6\frac{dy}{dx}-5y=0$ A) $y=Ae^x+Be^{5x}$. B) $y=Ae^{-x}+Be^{-5x}$. C) $y=Ae^{\left(-3+\sqrt[]{14}\right)x}+Be^{\left(-3-\sqrt[]{14}\right)x}$. D) $y=A\cos\left(-3+\sqrt[]{14}\right)x+B\sin\left(-3+\sqrt[]{14}\right)x$. Show Answer Correct Answer: C) $y=Ae^{\left(-3+\sqrt[]{14}\right)x}+Be^{\left(-3-\sqrt[]{14}\right)x}$. 21. $\frac{\partial^3z}{\partial x^3}-3\frac{\partial^3z}{\partial x^2\partial y}+4\frac{\partial^3z}{\partial y^3}=0$ A) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$. B) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y-2x\right)$. C) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y+2x\right)$. D) $Z=f_1\left(y+x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$. Show Answer Correct Answer: A) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$. 22. Find dy/dx if y = sec(8x$^{2}$). A) Sec(16x)tan(16x). B) 16xcos(8x$^{2}$). C) 16x$^{}$secxtanx(8x$^{2}$). D) 16xsec(8x$^{2}$)tan(8x$^{2}$). Show Answer Correct Answer: D) 16xsec(8x$^{2}$)tan(8x$^{2}$). 23. L$^{-1 }$\{s/(s$^{2}$+b$^{2}$)\} A) 1. B) T$^{n}$. C) E$^{at}$. D) T$^{n }$e$^{at}$. E) Cos bt. Show Answer Correct Answer: E) Cos bt. 24. Let PDE u$_{xx }$+ u$_{yy }$= u$_{t}$, By method separation, we consider the solution A) U(x, y)=X(x)Y(y). B) U(x, y, t)=X(x)Y(y)T(t). C) U(x, t)=X(x)T(t). D) None of these. Show Answer Correct Answer: B) U(x, y, t)=X(x)Y(y)T(t). 25. The value of $\nabla\cdot\overrightarrow{r}$ A) 3. B) 0. C) 1. D) 2. Show Answer Correct Answer: A) 3. ← PreviousNext →Related QuizzesClass 12 Mathematics Chapter 9 Differential Equations Quiz 1Class 12 Mathematics Chapter 9 Differential Equations Quiz 2Class 12 Mathematics Chapter 9 Differential Equations Quiz 3Class 12 Mathematics Chapter 9 Differential Equations Quiz 4Class 12 Mathematics Chapter 9 Differential Equations Quiz 5Class 12 Mathematics Chapter 9 Differential Equations Quiz 6Class 12 Mathematics Chapter 9 Differential Equations Quiz 7Class 12 Mathematics Chapter 9 Differential Equations Quiz 8Class 12 Mathematics Chapter 9 Differential Equations Quiz 9Class 12 Mathematics Chapter 9 Differential Equations Quiz 10 🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books