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SrEstroncio
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I have been doing some self study on differential equations using Tom Apostol's Calculus Vol. 1. and I got stuck on a problem (problem 12, section 8.5, vol. 1).
Let K be a non zero constant. Suppose P and Q are continuous in an open interval I.
Let a\in I and b a real number. Let v=g(x) be the only solution to the initial value problem v' +kP(x)v=kQ(x) in I, with g(a)=b. If n\not= 1 and k=1-n, prove the function y=f(x) not identically zero in I, is a solution to y'+P(x)y=Q(x)y^n and {f(a)}^k=b in I, if and only if the k-th power of f is equal to g in I.
The general solution of a linear first order differential equation is probably used.
I have tried several manipulations and substitutions, but I am kinda lost. I would appreciate any help.
Homework Statement
Let K be a non zero constant. Suppose P and Q are continuous in an open interval I.
Let a\in I and b a real number. Let v=g(x) be the only solution to the initial value problem v' +kP(x)v=kQ(x) in I, with g(a)=b. If n\not= 1 and k=1-n, prove the function y=f(x) not identically zero in I, is a solution to y'+P(x)y=Q(x)y^n and {f(a)}^k=b in I, if and only if the k-th power of f is equal to g in I.
Homework Equations
The general solution of a linear first order differential equation is probably used.
The Attempt at a Solution
I have tried several manipulations and substitutions, but I am kinda lost. I would appreciate any help.
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