A solution of a differential equation is an expression for the dependent variable in terms of the independent one(s) which satisfies the relation. The general solution includes all possible solutions and typically includes arbitrary constants (in the case of an ODE) or arbitrary functions (in the case of a PDE.)

Verifying a Solution to a Differential Equation In algebra when we are told to solve, it means get “y” by itself on the left hand side and no “y” terms on the right hand side. If y = f(x) is a solution to a differential equation, then if we plug “y” into the equation, we get a true statement.

Step 1 Separate the variables by moving all the y terms to one side of the equation and all the x terms to the other side:

1. Multiply both sides by dx:dy = (1/y) dx. Multiply both sides by y: y dy = dx.
2. Put the integral sign in front:∫ y dy = ∫ dx. Integrate each side: (y2)/2 = x + C.
3. Multiply both sides by 2: y2 = 2(x + C)

Separation of variables is a method of solving ordinary and partial differential equations. It is especially useful in solving equations arising in mathematical physics, such as Laplace’s equation, the Helmholtz differential equation, and the Schrödinger equation.

What are the two methods used to find the type of PDEs? Explanation: Partial differential equations can be classified using their characteristic lines. These are located using either the Cramer’s method or the Eigenvalue method. 2.

The multiple integral is a type of definite integral extended to functions of more than one real variable—for example, f(x,y) f ( x , y ) or f(x,y,z) f ( x , y , z ) .

Using the formula for integration by parts Example Find ∫ x cosxdx. v = ∫ cos xdx = sin x. ∫ udvdx dx = uv − ∫ vdu dx dx : ∫ x cosxdx = x sin x − ∫ (sin x) · 1dx = xsin x + cosx + c where c is the constant of integration.

General rule for differentiation: The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. ddx[k⋅f(x)]=kddx[f(x)] The derivative of a sum is equal to the sum of the derivatives.