Exercise · ZW-INITIAL-VALUE-PROBLEM-009 · ★★★ Advanced
The solution of \(y'=y^2,\ y(0)=1\) is \(y=\frac{1}{1-x}\). What is the largest interval containing \(0\) where it is defined?
Choose one option.
Show answer
- B. \((-\infty,1)\)
Explanation
It fails at \(x=1\); the interval must contain the initial point \(x_0=0\), so it is \((-\infty,1)\).
Review the concept: Initial value problem
An initial value problem asks for the solution of a differential equation that, in addition, takes given values at a given point.
\[y'=f(x,y),\qquad y(x_0)=y_0\]
More practice problems: initial value problem
- ZW-INITIAL-VALUE-PROBLEM-001 Solve y'=2y,\ y(0)=3.
- ZW-INITIAL-VALUE-PROBLEM-002 Solve y'=2x,\ y(0)=5.
- ZW-INITIAL-VALUE-PROBLEM-003 The general solution of y'=-y is y=Ce^-x. If y(0)=4, what is C?
- ZW-INITIAL-VALUE-PROBLEM-004 Solve y'=cos x,\ y(0)=1.
- ZW-INITIAL-VALUE-PROBLEM-005 Which function solves y'=y,\ y(1)=e?
- ZW-INITIAL-VALUE-PROBLEM-006 If y'=3x^2 and y(1)=2, what is y(2)?
- ZW-INITIAL-VALUE-PROBLEM-007 If y'=-2y and y(0)=1, what is y(ln 2)?
- ZW-INITIAL-VALUE-PROBLEM-008 Solve y''=6x with y(0)=1 and y'(0)=2.
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