Solve Diff. Eq. Problem Using Substitution Method - Step by Step Guide

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SUMMARY

The discussion focuses on solving the differential equation y' = (y + t)² using the substitution method. By substituting x = at + by + c, the equation transforms into a separable form x' = a + bf(x). The key steps involve rewriting the substitution as x - at - c = by and differentiating with respect to the independent variable t. This approach leads to a clearer path for finding the general solution of the given differential equation.

PREREQUISITES
  • Understanding of differential equations, specifically first-order equations.
  • Familiarity with the substitution method in solving differential equations.
  • Knowledge of derivatives and their application in calculus.
  • Basic algebra skills for manipulating equations and expressions.
NEXT STEPS
  • Study the method of substitution in solving differential equations.
  • Learn about separable differential equations and their solutions.
  • Explore the concept of derivatives and their role in changing variables.
  • Practice solving various first-order differential equations using different methods.
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Students and professionals in mathematics, particularly those studying calculus and differential equations, as well as educators looking for effective teaching methods in these topics.

scienceman2k9
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Problem:

y'=f(at+by+c) where a,b, and c are constants. Show that the substitution of x=at+by+c changes the equation to the separable equation x'=a+bf(x). Use this method to find the general solution of the equation y'=(y+t)^2


I really have no clue on this one. If someone could start me off in the right direction that would be great. I'm not seeing how they get that separable eq.
 
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Try writting the substitution as
[tex]x-at-c = by[/tex]
Now take the derivative with respect to what I am assuming is the independent variable, t. The result should pop out at you.
 

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