Calculus

First Order Differential Equations

Differential Equations - Degree and Order

         

Which of the following differential equations is a linear equation of order 3:3: (a)d3ydx3+d2ydx2dydx+y=5x(b)d3ydx3+d2ydx2+y2=x2+3x(c)xd3ydx3+d2ydx2=ex(d)d2ydx2+dydx=logx?\begin{aligned} &(a) \frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}\frac{dy}{dx}+y=5x\\ &(b) \frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}+y^2=x^2+3x\\ &(c) x\frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}=e^x\\ &(d) \frac{d^2y}{dx^2}+\frac{dy}{dx}=\log{x}? \end{aligned}

If aa and bb are the order and the degree, respectively, of the differential equation (d4ydx4)8(dydx)25+15=0,\left(\frac{d^{4}y}{dx^{4}}\right)^{8}-\left(\frac{dy}{dx}\right)^{25}+15=0, what is the value of ba?b-a?

What is the degree of the differential equation d3ydx3+12(d2ydx2)2=x3logd2ydx2?\frac{d^3y}{dx^3}+12\left(\frac{d^2y}{dx^2}\right)^2=x^3\log\frac{d^2y}{dx^2} ?

What is the degree of the following ordinary differential equation: k(y)5=(1+(y)4)4?k(y'')^{5}=(1+(y'')^{4})^{4}?

Which of the following differential equations has degree 1?1 ? (a) x3d2ydx2+(x+x2)(dydx)2+exy3=2sinx(b) y=2dydx+51(dydx)2(c) dydx+y=x+4\begin{aligned} &\text{(a) } x^3\frac{d^2y}{dx^2}+(x+x^2)\left(\frac{dy}{dx}\right)^2+e^xy^3=2\sin x \\ &\text{(b) } y=2\frac{dy}{dx}+5\sqrt{1-\left(\frac{dy}{dx}\right)^2} \\ &\text{(c) } \sqrt{\frac{dy}{dx}+y}=x+4 \end{aligned}

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