AP Calculus BC 7.2 Verifying Solutions for Differential Equations Questions

Review differential-equation verification by differentiating a proposed function and substituting it into the equation, including checking any stated domain.

Covered syllabus points

FUN-7.B—Verify solutions to differential equations

[Maximum number: 3]

Consider the differential equation $\frac{d y}{d x}=2 x-y$ .

Find the values of the constants m and b for which y=m x+b is a solution to the differential equation.

$y=m x+b \Rightarrow \frac{d y}{d x}=\frac{d}{d x}(m x+b)=m$

\begin{aligned} & 2 x-y=m \\ & 2 x-(m x+b)=m \\ & (2-m) x-(m+b)=0 \\ & 2-m=0 \Rightarrow m=2 \\ & b=-m \Rightarrow b=-2 \end{aligned}

Therefore, m=2 and b=-2.

2:\arrayl1: slopes where x=0
1: slopes where x=1array.

2:\arrayl1: d^2 yd x^2
1: concave up with reason array.

2:\arrayl1: considers.d y/d x|_(x, y)=(2,3)
1: conclusion with justification array.

3:\arrayl1: d/d x(m x+b)=m
1: 2 x-y=m
1: answer array.

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Covered syllabus points

Question 1