It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation – (what is the highest number of derivatives involved) and whether or not the equation is linear. Linearity is important because the structure of the the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly. Determine whether or not each equation is linear: 1. dy/dt + ty" = 0 2. d?y/dt? + d3y/dt3 + d"y/dt" + dy/dt = 1 3. d3y/dt3 + t(dy/dt) + (cos"(t))y = t3 4. d"y/dt" + sin(t + y) = sin t
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A linear differential equation is defined as an equation where the dependent variable and its derivatives appear in a linear manner, i.e., they are not multiplied together, raised to any power other than 1, or used as arguments in non-linear functions. Show more…
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Determine the order of the given differential equation; also state whether the equation is linear or nonlinear. $$ \frac{d^{2} y}{d t^{2}}+\sin (t+y)=\sin t $$
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Classification of DEs If a differential equation can be written as a linear combination of the unknown function and its derivatives, it is called linear. Note that the coefficients may be functions of the independent variable. Here is the most general form of a linear ODE of order n: ao(t)y(t) + a1(t)y'(t) + ... + an(t)y^(n)(t) = g(t) Otherwise, the DE is called nonlinear. Example 5. Classify each DE as an ordinary or partial, linear or nonlinear, and indicate its order. a) y'' + = cos(t) b) y'' + (y^2 - 1)y' + t^3y tan(t) c) u'' + 3u'v + (2 + y^3)u = 1 + u d) u''r + u^2y = 0
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$1-4$ Determine whether the differential equation is linear. $$y^{\prime}=\frac{1}{x}+\frac{1}{y}$$
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