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Lesson 5a - Linear - y a x y b x I x...

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Linear First Order ODEs C id fi t d ODE th t l k lik th f ll i ) ( ) ( ' b Consider a first order ODE that looks like the following: x y x a y If we introduce a function called and multiply this through the ) ( x I equation, we will get the following ODE instead, where would be the function that makes: ) ( x I y x a x I y x I y x I ) ( ) ( ' ) ( ]' ) ( [ By the product rule, we know that y x I y x I y x I ) ( ' ' ) ( ]' ) ( [ Can we figure out ? ) ( x I Combining these two ideas we get: y x a x I y x I y x a x I y x I ) ( ) ( ' ' ) ( ) ( ) ( ' ) ( y x a x I y x a x I ) ( ) ( ' ) ( ) ( ) ( ' ) ( ) ( I I dx x I du x I u ) ( ' then ), ( Let 1 x x a x ) ( ) ( ' ) ( x I x I x a du u dx x a ) ( | | ln ) ( u dx x a dx x I x I dx x a ) ( ) ( ' ) ( | ) ( | ln ) ( x I dx x a
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Linear Idea Continued | ) ( | l ) ( I d )| ( | l ) ( d ) ( d ln x dx x a ln x I dx x a e e ) ( x I e dx x a So we are able to find but why is this important? W ll b h d fi d t k ) ( x I I I I ) ( ) ( ' ) ( ]' ) ( [ ) ( I Well… remember how we defined our to work: ) ( ) ( ' x b y x a y ) ( ) ( ) ( ) ( ' ) ( x b x I y x a x I y x I y x a x y x y x ) ( ) ( ]' ) ( [ x b x I y x I x ] dx x
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