Chapter 4: Differential Equations January 14, 2025 Maven Leave a Comment Welcome to your Chapter 4: Differential Equations 1. The degree of the differential equation $ \dfrac{d^{4}y}{dx^{4}}- \bigg (\dfrac{d^{2}y}{dx^{2}}\bigg)^{4}+\dfrac{dy}{dx}=3 $ $1$ $2$ $3$ $4$ None 2. The integrating factor of the differential equation $\dfrac{dx}{dy}+Px= Q$ is $e^{\int_{}^{}Pdx}$ ${\int_{}^{}Pdx}$ ${\int_{}^{}Pdy}$ $e^{\int_{}^{}Pdy}$ None 3. If $ y = cx + c − c^3$ then its differential equation is $y= x\frac{dy}{dx}+ \frac{dy}{dx}-\bigg(\frac{dy}{dx}\bigg)^{3}$ $y+ \bigg(\frac{dy}{dx}\bigg)^{3}= x\frac{dy}{dx}- \frac{dy}{dx}$ $\frac{dy}{dx}+ y= \bigg(\frac{dy}{dx}\bigg)^{3}- x\frac{dy}{dx}$ $\frac{d^{3}y}{dx^{3}}= 0$ None 4. The differential equation $\bigg(\dfrac{dx}{dy}\bigg)^{{3}} +2y^{\dfrac{1}{2}}=x\; is $ of order 2 and degree 1 of order 1 and degree 3 of order 1 and degree 6 of order 1 and degree 2 None 5. The differential equation formed by eliminating a and b from $y=ae^{x}+be^{-x}$ is $\frac{d^{2}y}{dx^{2}}-y=0 $ $\frac{d^{2}y}{dx^{2}}-\frac{dy}{dx}=0 $ $\frac{d^{2}y}{dx^{2}}=0 $ $\frac{d^{2}y}{dx^{2}}-x=0 $ None 6. The complementary function of $(D^{2}+4)y=e^{2x}$ is $(Ax+B)e^{2x}$ $(Ax+B)e^{-2x}$ $Acos2x + Bsin2x$ $Ae^{-2x}+Be^{2x}$ None 7. The order and degree of the differential equation $ \sqrt{\dfrac{d^{2}y}{dx^{2}}} = \sqrt{\dfrac{dy}{dx}+5} $ are respectively 2 and 3 3 and 2 2 and 1 2 and 2 None 8. The differential equation of y = mx + c is (m and c are arbitrary constants) $\frac{d^{2}y}{dx^{2}}=0$ $y=x\frac{dy}{dx}+c$ $xdy + ydx = 0$ $ydx − xdy = 0$ None 9. The particular integral of the differential equation $\dfrac{d^{2}y}{dx^{2}}-8\dfrac{dy}{dx}+16y=2e^{4x}$ is $\frac{x^{2}e^{4}x}{2!}$ $\frac{e^{4}x}{2!}$ ${x^{2}e^{4}x}$ ${xe^{4}x}$ None 10. The order and degree of the differential equation $\bigg(\dfrac{d^{2}y}{dx^{2}}\bigg)^{\dfrac{3}{2}}-\sqrt{\dfrac{dy}{dx}}-4=0 $ are respectively 2 and 6 3 and 6 1 and 4 2 and 4 None 11. A homogeneous differential equation of the form $\dfrac{dx}{dy}=f(\dfrac{x}{y})$ can be solved by making substitution, $x = v y$ $y = v x$ $y = v$ $x = v$ None 12. Which of the following is the homogeneous differential equation? $(3x − 5) dx = (4y −1) dy$ $xy dx − (x^3 + y^3 ) dy = 0$ $y^2 dx + (x^2 − xy − y^2 ) dy = 0$ $(x^2 + y) dx =(y^2 + x) dy$ None 13. A homogeneous differential equation of the form $\dfrac{dy}{dx}=f(\dfrac{y}{x})$ can be solved by making substitution, $y = v x$ $v = y x$ $x = v y$ $x = v$ None 14. The variable separable form of $ \dfrac{dy}{dx}=\dfrac{y(x-y)}{x(x+y)}$ by taking y vx and $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$ $\frac{2v^{2}}{1+v}dv=\frac{dx}{x}$ $\frac{2v^{2}}{1+v}dv=\frac{-dx}{x}$ $\frac{2v^{2}}{1-v}dv=\frac{-dx}{x}$ $\frac{1+v}{2v^{2}}dv=\frac{-dx}{x}$ None 15. The solution of the differential equation $\dfrac{dy}{dx}=\dfrac{y}{x}+\dfrac{f(\dfrac{y}{x})}{f'(\dfrac{y}{x})}$ is $ f\bigg(\frac{y}{x}\bigg)=kx$ $ xf\bigg(\frac{y}{x}\bigg)=k$ $ f\bigg(\frac{y}{x}\bigg)=ky$ $ yf\bigg(\frac{y}{x}\bigg)=k$ None 16. The integrating factor of $x \dfrac{dy}{dx}-y=x^{2}$ is $\frac{-1}{x}$ $\frac{1}{x}$ logx x None 17. The differential equation of $x^2 + y^2 = a^2$ $xdy+ydx=0$ $ydx–xdy=0$ $xdx–ydx=0$ $xdx+ydy=0$ None 18. The particular integral of the differential equation $f(D)y =e^{ax}$ where $ f(D)=(D-a)^{2}$ $\frac{x^{2}}{2}e^{ax}$ $xe^{ax}$ $\frac{x}{2}e^{ax}$ $x^{2}e^{ax}$ None 19. The P.I of $(3D^2 + D − 14)y =13e^2x$ is $\frac{x}{2} e^{2x}$ ${x} e^{2x}$ $\frac{x^{2}}{2} e^{2x}$ ${13x} e^{2x}$ None 20. The differential equation formed by eliminating A and B from $y = e^{−2x} (Acos x + Bsin x)$ is $y_2 − 4y_1 + 5 = 0$ $y_2+4y - 5 = 0$ $y_2-4y_1 - 5 = 0$ $y_2+4y_1 -+5 = 0$ None 21. The general solution of the differential equation $\dfrac{dy}{dx}=cosx$ is $y = sin x +1$ $y = sin x - 2$ $y = cos x + c, $ c is an arbitrary constant $y = sin x + c, $ c is an arbitrary constant None 22. The solution of the differential equation $\dfrac{dy}{dx}+Py=Q$ where P and Q are the function of x is $y=\int_{}^{}Qe^{\int_{}^{}Pdx}dx+c$ $y=\int_{}^{}Qe^{\int_{}^{}-Pdx}dx+c$ $y e^{\int_{}^{}Pdx}=\int_{}^{}Qe^{\int_{}^{}Pdx}dx+c$ $y e^{\int_{}^{}Pdx}=\int_{}^{}Qe^{\int_{}^{}-Pdx}dx+C$ None 23. The complementary function of $\dfrac{d^{2}y}{dx^{2}}-\dfrac{dy}{dx}=0$ is $A+Be^{x}$ $(A+B)e^{x}$ $(Ax+B)e^{x}$ $(Ae^x+B)$ None 24. Solution of $\dfrac{dx}{dy}+Px=0$ $x=ce^{py}$ $x=ce^{-py}$ $x=py+c$ $x=cy$ None 25. If $sec^{2}x$ is an integrating factor of the differential equation $\dfrac{dy}{dx}+Py=Q$ then P= $2 tan x$ $sec x$ $cos^{2}x$ $tan^{2}x$ None Time's up Related Posts:Chapter 5: Differential CalculusChapter 1: Introduction to Micro EconomicsChapter 2. Consumption AnalysisChapter 3: Production Analysis
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