Exercise · ZW-SOLUTION-OF-A-DIFFERENTIAL-EQUATION-007 · ★★☆ Intermediate
For which value of \(a\) is \(y=ax^2\) a solution of \(y'=x\)?
Choose one option.
Show answer
- A. \(\tfrac12\)
Explanation
Since \(y'=2ax\), we need \(2ax=x\), that is \(a=\tfrac12\).
Review the concept: Solution of a differential equation
A solution \(\varphi\) of a differential equation \(F=0\) is a function that, substituted together with its derivatives, makes the equation a true identity on a whole interval.
\[F\big(x,\varphi(x),\varphi'(x),\dots,\varphi^{(n)}(x)\big)=0\]
More practice problems: solution of a differential equation
- ZW-SOLUTION-OF-A-DIFFERENTIAL-EQUATION-001 Which of these functions is a solution of y'=3y?
- ZW-SOLUTION-OF-A-DIFFERENTIAL-EQUATION-002 Which is a solution of y''+y=0?
- ZW-SOLUTION-OF-A-DIFFERENTIAL-EQUATION-003 The function y=x^3 is a solution of xy'=ky. What is k?
- ZW-SOLUTION-OF-A-DIFFERENTIAL-EQUATION-004 What can be said about y=x^2+C with respect to y'=2x?
- ZW-SOLUTION-OF-A-DIFFERENTIAL-EQUATION-005 For which value of k is y=e^kx a solution of y''-5y'+6y=0?
- ZW-SOLUTION-OF-A-DIFFERENTIAL-EQUATION-006 For which value of k is y=sin (kx) a solution of y''+9y=0?
- ZW-SOLUTION-OF-A-DIFFERENTIAL-EQUATION-008 Which equation does y=xln x (with x>0) satisfy?
- ZW-SOLUTION-OF-A-DIFFERENTIAL-EQUATION-009 The function y=(1)/(1-x) satisfies y'=y^2. On which interval is it a solution?
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