Differential Equations

Definition

A differential equation is an equation containing derivatives of a dependent variable \(y\) with respect to independent variables \(x\). In particular,

• Ordinary Differential Equations (ODE) are differential equations having one independent variable.

• Partial Differential Equations (PDE) are differential equations having two or more independent variables.

Order

An ODE is said to be of order \(n\) if the highest derivative of the unknown function in the equation is the \(n\)th derivative with respect to the independent variable.

Linearity

\(x\)\(y\)\(n\)\(y^{(n)}\)\(x\)\(b(x)\)\(a_i(x)\) and all of its derivatives appear by themselves. Thus, it is of the form: An ODE is said to be linear only if the function

First Order Differential Equations

1. \[\int A(x)dx + \int B(y)dy = C;\]
   , it is separable, and the solution follows by integration: If the equation can be put in the form

2. If \(M(x,y)\) and \(N(x,y)\) are homogeneous and of the same degree in \(x\) and \(y\), then substitution of \(vx\) for \(y\) (thus, \(dy = v dx + x dv\)) will yield a separable equation in the variables \(x\) and \(y\). (A function such as \(M(x,y)\) is homogeneous of degree \(n\) in \(x\) and \(y\) if \(M(cx,cy) = c^n M(x,y)\)).

3. If \(M(x,y)dx + N(x,y)dy\) is the differential of some function \(F(x,y)\), then the given equation is said to be exact. A necessary and sufficient condition for exactness is \(\frac{dM}{dy} - \frac{dN}{dx}\). When the equation is exact, \(F\) is found from the relations \(\frac{\partial F}{\partial x} = M\) and \(\frac{\partial F}{\partial y} = N\), and the solution is \(F(x,y) = C\) (a constant).

4. \[\frac{dy}{dx} + P(x)y = Q(x).\]
   yields . Multiplication by : Such an equation has the form Linear, order one in

Second Order Linear Equations (with Constant Coefficients)

1. Right-hand side: 0 (homogeneous case)

   The second-order linear homogeneous equation is given by:

   \[(b_2D^2 + b_1D + b_0)y = 0.\]

   The auxiliary equation associated with this equation is:

   \[b_2m^2 + b_1m + b_0 = 0.\]

   If the roots of the auxiliary equation are real and distinct, denoted as \(m_1\) and \(m_2\), the solution is:

   \[y = C_1e^{m_1x} + C_2e^{m_2x},\]

   where \(C_1\) and \(C_2\) are arbitrary constants.

   If the roots of the auxiliary equation are real and repeated, say \(m_1 = m_2 = p\), the solution is:

   \[y = C_1e^{px} + C_2xe^{px}.\]

   If the roots of the auxiliary equation are complex, \(a + ib\) and \(a - ib\), the solution is:

   \[y = C_1e^{ax}\cos(bx) + C_2e^{ax}\sin(bx).\]

2. Right-hand side: \(f(x)\) (nonhomogeneous case)

   The second-order linear nonhomogeneous equation is given by:

   \[(b_2D^2 + b_1D + b_0)y = f(x).\]

   The general solution is:

   \[y = C_1y_1(x) + C_2y_2(x) + y_p(x),\]

   where \(y_1(x)\) and \(y_2(x)\) are solutions of the corresponding homogeneous equation, and \(y_p(x)\) is a particular solution of the given nonhomogeneous equation. The particular solution \(y_p(x)\) has the form:

   \[y_p(x) = A(x)y_1(x) + B(x)y_2(x),\]

   and \(A(x)\) and \(B(x)\) are found from the simultaneous solution of \(A'y_1 - B'y_2 = 0\) and \(f(x)/b_2\). A solution exists if the determinant

   \[\begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix}\]

   does not equal zero. The simultaneous equations yield \(A'\) and \(B'\) from which \(A\) and \(B\) follow by integration.