Exercise · ZW-FIRST-ORDER-LINEAR-EQUATION-009 · ★★★ Advanced
The general solution of \(y'+y=x\) is \(y=x-1+Ce^{-x}\). Which is the solution with \(y(0)=0\)?
Choose one option.
Show answer
- D. \(y=x-1+e^{-x}\)
Explanation
\(y(0)=-1+C=0\) gives \(C=1\). The other options equal \(-1\) or \(-2\) at \(x=0\), or (\(x+1-e^{-x}\)) equal \(0\) but are not of the form \(x-1+Ce^{-x}\).
Review the concept: First-order linear equation
A first-order linear equation is a differential equation in which \(y\) and \(y'\) appear to the first power, with a coefficient \(p\) and a free term \(q\) that depend only on \(x\).
\[y'+p(x)\,y=q(x)\]
More practice problems: first-order linear equation
- ZW-FIRST-ORDER-LINEAR-EQUATION-001 Which of these equations is a first-order linear equation?
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- ZW-FIRST-ORDER-LINEAR-EQUATION-004 The equation x y'+2y=x^2, with x>0, is brought to standard form by dividing by x. What is p(x)?
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- ZW-FIRST-ORDER-LINEAR-EQUATION-006 Find the general solution of y'+2xy=0.
- ZW-FIRST-ORDER-LINEAR-EQUATION-007 Solve y'+y=0, y(0)=5.
- ZW-FIRST-ORDER-LINEAR-EQUATION-008 Which of these functions is a solution of y'+y=x?
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