## Higher-Order Differential Equations

Here, 1( ), 2( ), ii , ( ) make up a fundamental set of solutions (the

are linearly independent solutions;

ƒ‚

0); ﬂ 1, ﬂ 2, ii , ﬂ are arbitrary constants. 2

. Let 0 = 0( ) be a nontrivial particular solution of equation (1). Then the substitution

= 0( ) „…E ( ) 67

leads to a linear equation of the ( O − 1)st-order for E ( ). Let 1 = 1( ) and 2 = 2( ) be two nontrivial linearly independent particular solutions of

equation (1) with / ≡ 0. Then the substitution

leads to a linear equation of the ( O − 2)nd-order for = ( ). 3

. Some additional information about higher-order linear equations can be found in Subsections 0.4.1-0.4.3.