MATH 222 – MIDTERM 2 MASTERY GUIDE | DIFFERENTIAL EQUATIONS, SEQUENCES, SERIES & POWER SERIES

ESTUDYR


MATH 222 – MIDTERM 2 MASTERY GUIDE | DIFFERENTIAL
EQUATIONS, SEQUENCES, SERIES & POWER SERIES
1) What is the correct general process for solving a separable differential equation for yyy?

A. Integrate yyy side only, then substitute xxx
B. Solve for xxx first, then differentiate
C. Separate variables → integrate both sides (+C) → isolate yyy
D. Use integrating factor

Rationale: Separable DEs require rewriting as g(y) dy=f(x) dxg(y)\,dy = f(x)\,dxg(y)dy=f(x)dx,
integrating both sides, then solving for yyy.



2) Which tactic is most commonly required when integrating after separating variables?

A. Only geometric series
B. Only logarithm rules
C. Substitution, integration by parts, or partial fractions
D. Laplace transforms only

Rationale: After separation, integrals can be nontrivial and may require standard integration
techniques.



3) When solving a separable differential equation, why is it useful to rewrite y′y'y′ as
dydx\frac{dy}{dx}dxdy?

A. To avoid constants
B. To treat dydydy and dxdxdx as separable differentials
C. To convert into a power series
D. To apply comparison tests

Rationale: Separation requires isolating yyy-terms with dydydy and xxx-terms with dxdxdx.



4) For a separable DE with an initial condition, what extra step is required?

A. Find the radius of convergence
B. Apply the divergence test

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C. Use the initial condition to solve for the constant CCC
D. Convert to a geometric series

Rationale: The initial condition pins down the particular solution by determining CCC.



5) A proposed solution to an IVP must satisfy which conditions?

A. Only the initial condition
B. Only the differential equation
C. Both the differential equation and the initial condition
D. Neither—only continuity matters

Rationale: An IVP solution must be a solution to the DE and match the initial value.



6) If a function satisfies the differential equation but not the initial condition, it is:

A. Not a solution at all
B. A solution of the DE, but not of the IVP
C. A solution of the IVP only
D. Automatically invalid

Rationale: The DE can have a family of solutions; the IVP selects one.



7) Which is a correct property of natural logarithms?

A. ln⁡(x+y)=ln⁡x+ln⁡y\ln(x+y)=\ln x+\ln yln(x+y)=lnx+lny
B. ln⁡(x/y)=ln⁡x+ln⁡y\ln(x/y)=\ln x+\ln yln(x/y)=lnx+lny
C. ln⁡(xy)=ln⁡x+ln⁡y\ln(xy)=\ln x+\ln yln(xy)=lnx+lny
D. ln⁡(xa)=ln⁡xa\ln(x^a)=\ln x^aln(xa)=lnxa (no simplification)

Rationale: Log rules: product becomes sum.



8) Which is the correct quotient rule for logs?

A. ln⁡(x/y)=ln⁡x/ln⁡y\ln(x/y)=\ln x / \ln yln(x/y)=lnx/lny
B. ln⁡(x/y)=ln⁡(x+y)\ln(x/y)=\ln(x+y)ln(x/y)=ln(x+y)

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C. ln⁡(x/y)=ln⁡x−ln⁡y\ln(x/y)=\ln x-\ln yln(x/y)=lnx−lny
D. ln⁡(x/y)=ln⁡y−ln⁡x\ln(x/y)=\ln y-\ln xln(x/y)=lny−lnx

Rationale: Dividing inside a log becomes subtraction.



9) Which is the correct power rule for logs?

A. ln⁡(xa)=ln⁡x+a\ln(x^a)=\ln x + aln(xa)=lnx+a
B. ln⁡(xa)=ln⁡x/a\ln(x^a)=\ln x/aln(xa)=lnx/a
C. ln⁡(xa)=aln⁡x\ln(x^a)=a\ln xln(xa)=alnx
D. ln⁡(xa)=ax\ln(x^a)=a^xln(xa)=ax

Rationale: Exponent becomes a multiplier.



10) A first-order linear differential equation can be written as:

A. y′′+P(x)y′=Q(x)y'' + P(x)y' = Q(x)y′′+P(x)y′=Q(x)
B. y′P(x)+yQ(x)=0y'P(x) + yQ(x) = 0y′P(x)+yQ(x)=0
C. dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x)dxdy+P(x)y=Q(x)
D. y′=y2+xy' = y^2 + xy′=y2+x

Rationale: This is the standard linear first-order form.



11) What is the first step when solving a linear DE using integrating factors?

A. Take a limit
B. Verify/rewrite into y′+P(x)y=Q(x)y' + P(x)y = Q(x)y′+P(x)y=Q(x)
C. Separate variables
D. Convert to a geometric series

Rationale: Integrating factors only apply once the DE is in standard linear form.



12) The integrating factor for y′+P(x)y=Q(x)y' + P(x)y = Q(x)y′+P(x)y=Q(x) is:

A. I(x)=∫P(x) dxI(x)=\int P(x)\,dxI(x)=∫P(x)dx
B. I(x)=ln⁡(P(x))I(x)=\ln(P(x))I(x)=ln(P(x))