## A Short Course on Differential Equations |

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algebraic expression arbitrary constants arbitrary function auxiliary equation CHAPTER circuit constant and equal constant coefficients continuous function cosec Definition denotes Determine the curve differentiation with respect dv dv dv dx dx dy dx dy dz dy dx dy dz dz dz dz dx Eliminate equa equation becomes equation Mdx f Example following equations following exercises g0 is arbitrary hand member zero independent variable indicial equation infinite series Laplace's Equation left hand member Legendre's Equation linear differential equation Multiply necessary and sufficient ordinary differential equation ordinary linear differential original equation partial differential equation partial fractions particular solutions positive integer power series preceding article regular point right hand member roots satisfies the equation second order single valued solu Solve the equation spherical coordinates Substitute subtangent Suppose symbolic factors theorem tion valued and continuous

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Page 59 - E is the modulus of elasticity, / is the moment of inertia of a cross section of the beam about a gravity axis in the section perpendicular to the applied forces, and I is the length of the beam.

Page 22 - Mdx + Ndy = 0 where M and N are functions of x and y and do not contain derivatives.

Page 7 - ... 7. In examples (1) to (4) inclusive of the preceding article it will be noticed that differentials enter the equation only in derivatives. It is conceivable, however, that there might be an equation containing differentials other than those in the derivatives, as for example, but there is no need of entering into a discussion of such equations, and we shall not do so. In what follows, we shall assume that if the equation is written in differential form, the differentials can all be converted...

Page 53 - The result when the operator is applied to the sum of a number of functions is equal to the sum of the results found when the operator is applied to each of the functions separately.

Page 25 - Integrate each term separately, and write the sum of their integrals equal to an arbitrary constant. (2) M and N homogeneous functions of x and y of the ~same degree. Introduce in place of...

Page 36 - Plot the curve when n = \. 45. Determine the curve in which the polar subnormal is proportional to the sine of the vectorial angle. 46. Determine the curve in which the polar subtangent is proportional to the length of the radius vector. The equation for a circuit containing induction and resistance is...

Page 29 - FACTORS It sometimes happens that the differential equation Mdx + Ndy = 0 is not exact but becomes so when it is multiplied by some quantity. Thus, of Art. 21, is not exact but becomes so after multiplication by effdx. Definition. A factor which changes a differential equation into an exact differential equation is called an integrating factor of the equation. , Sometimes an integrating factor can be found by inspection. EXAMPLE. Find the general solution of the equation (zV _ y*)dx + 2xydy = 0.

Page 26 - Ndy = 0 where M and N are functions of x and y, is said to be exact when there is a function u(x, y) such that du = Mdx -f- Ndy.

Page 9 - A solution of an ordinary differential equation may be one of three kinds: general, particular and singular. A general solution is one which contains arbitrary constants equal in number to the exponent of the order of the equation. Thus, in example 1, Art. 10, the number of arbitrary constants is one and the exponent of the order of the equation is 1, and in example 2 of the same article the number of arbitrary constants is two, and the exponent of the order of the equation is 2. In either case the...