MATH 202 Notes, Summer 2002

Differential Equations Equation Method Take , then multiply the differential equation by I and write it down in the form then integrate Bernoulli’s Equation (E): with Substitute then differentiate y in terms of u and replace y and y’ in (E) to get (E’): If differential equation can be written in the form Substitute y = ux or x = yv , then transform the resulting equation into a separable equation If the differential equation (E): is exact differential equation i.e. Note: If it is written as rewrite it as same as where and Take . Then do the following 1. 2. and evaluate 3. evaluate and plug it in 4. The implicit solution is F = c If the differential equation (E): is not exact, turn it into exact by multiplying it with a specific integrating factor 1. If is a function of x alone, then multiply (E) by 2. If is a function of y alone, then multiply (E) by Reduction of order can be used to find the general solution of a nonhomogeneous DE whenever a solution is known can be obtained directly by: Homogeneous equation of the form with constant coefficients Find the roots of characteristic equation  If roots are distinct and real then the solution is  If roots are repeated and real then the solution is  If roots are complex i.e. Then the solution is Nonhomogeneous equation of the form The solution has the form where is the solution of associated homogeneous equation. Find by method of variation of parameters or undetermined coefficients To solve using variation of parameters divide by to get the 2nd order DE and find To find , integrate and to get & where W is the Wronskian , , and 1. Any constant 2. 3. 4. 5. 6. 7. 8. 9. 10. mix Forms of 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Cauchy-Euler equation of the form Find the roots of auxiliary equation  If roots and are real and distinct then the solution is  If the roots are real and equal i.e. , then the solution is  If the roots are the conjugate pair and then Cauchy-Euler DE with cte coefficients Substitute differentiate w.r.t. t When a linear DE has variable coefficients, find a solution in the form of infinite series Use Power series method where Definition of Laplace of f(t) = F(s) Laplace of a power function Laplace of the function, whose denominator can be written in factors. Use partial fractions Translation Theorem Convolution of two functions Convolution Theorem Differentiation of Transforms Integration of transforms To solve the systems of first order linear differential equations Write down in matrix form and use eigen value method Eigen values To find eigen values solve Eigen vectors To find an eigen vector, find V such that If eigen values are distinct, then the solution is If eigen values are repeated and you can’t find linearly independent eigen vectors If the eigen values are complex Find the eigen vector V for Take then separate its real and imaginary parts The solution is a linear combination of the real and imaginary parts