Which is the required differential equation of the given family of curves. SOLUTION The equation of the given family of curves is y = eˣ(A cos X + B sin x)ĭy/dx =eˣ(- A sin x + B cos x) + eˣ(A cosx + B sin x)ĭ²y/dx²-dy/dx = eˣ (- A cos x – B sin x) + eˣ (- A sin x + B cosx )ĭ²y/dx²-dy/dx=y(dy/dx-) Which is the required differential equation.Ģ- Find the differential equation of the family of curves y=eˣ(A cos x + B sin x), where A and B are arbitrary constants. Since the given equation contains two arbitrary constants, we differentiate it two times w.r.t. SOLUTION The equation of the given family of curves is y = Aeˣ + Be⁻ˣ. This gives us the required differential equation of order n.ġ – Find the differential equation of the family of curves y = Aeˣ + Be⁻ˣ, where A and B are arbitrary constants. Eliminate n constants, using these (n + 1) equations. Differentiate the equation of the given family of curves n times to get n more equations. Then, we obtain its differential equation, as given below. This is an example of an ODE of order mwhere mis a highest order of the derivative in the equation. Systems (j)General systems, Cauchy problem, existence and uniqueness. Euler’s equations: reduction to equation with constant coe cients.
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Suppose an equation of a family of curves contains n arbitrary constants (called parameters). be downloadedTextbook in pdf formatandTeX Source(when those are. Hence,y=Acosx-Bsinx is solution of the differentiol equation d²y/dx²+y=0 Verify that y= A cos x – B sin x is a solution of the differential equation a PARTICULAR SOLUTION OF A DIFFERENTIAL EQUATION Giving particular values to arbitrary constants in the general solution of a differential equation, we get its particular solutions. If its solution contains n arbitrary constants then it is called a general solution. Suppose a differential equation of order n is being given. GENERAL SOLUTION OF A DIFFERENTIAL EQUATION So, it is a differential equation of order 1 and degree 1.Ī function of the form y=f(x) +C which satisfies a given differential equation is called its solution. (iv) The equation y dx = x dy may be written as dy/dx=y/x. (iii) The equation x(d²y/dx²)³ + y(dy/dx)⁴ = 0 is an equation of order and degree 3. Hence, the above equation is of order 2 and degree 1. The power of the highest-order derivative is 1. In this equation, the order of the highest-order derivative is 2. (ii) Consider the equation dy/dx² +3(dy/dx)³+2y=0 Hence, the above equation is of order 1 and degree 2. The power of the highest-order derivative is 2. In this equation, the order of the highest-order derivative is 1. (i) Consider the equation (dy/dx)² + 5y = sin x.
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The power of the highest-order derivative occurring in a differential equation, after it is made free from radicals and fractions, is called the degree of the differential equation The order of the highest-order derivative occurring in a differential equation is called the order of the differential equation. An equation containing an independent variable, a dependent variable and the derivatives of the dependent variable is called a differential equation.Įxamples Each of the following equations is differential equation:
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