Solve y′=3y with y(0)=2.
Answer and explanation
y(t)=2e^(3t).
Credit a general exponential solution, applying y(0), and correct substitution.
Subjects
Differential equations
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Sample exercises
For y′+2y=8, find the equilibrium solution value.
Answer and explanation
4
Set y′=0 so 2y=8.
Solve y′+2y=8 with y(0)=1.
Answer and explanation
y(t)=4−3e^(−2t).
Credit the equilibrium plus a decaying homogeneous solution and the initial condition.
For y′=−5y, what happens to small deviations from y=0 as t increases?
- They become instantly zero
- They decay exponentially
- They grow exponentially
- They oscillate without decay
Answer and explanation
They decay exponentially
The solution is Ce^(−5t), so the equilibrium is asymptotically stable.
For y′=2t and y(0)=3, calculate y(2).
Answer and explanation
7
Integration gives y=t²+3, so y(2)=7.
Why does checking only the initial condition fail to verify a proposed solution?
Answer and explanation
Infinitely many functions pass through the initial point. The function must also satisfy the differential equation throughout its interval.
Credit the distinction between a pointwise constraint and a differential relation.
Study notes
Initial-value problems and equilibrium
Solving a differential equation produces constants; initial conditions select a particular solution.
UniversityInitial-value problems and equilibrium · revision notes
Solving a differential equation produces constants; initial conditions select a particular solution.
UniversitySecond-order equations and resonance
For constant coefficients, an exponential trial function reduces the homogeneous ODE to a polynomial.
UniversitySecond-order equations and resonance · revision notes
For constant coefficients, an exponential trial function reduces the homogeneous ODE to a polynomial.
UniversityFirst-order exponential initial-value problem
Use the initial condition after solving the homogeneous differential equation.
UniversityFirst-order exponential initial-value problem · formula reference
Use the initial condition after solving the homogeneous differential equation.
UniversityA constant-forced first-order equation
For y′ + ay = b with constant a > 0, b/a is the equilibrium value.
UniversityA constant-forced first-order equation · formula reference
For y′ + ay = b with constant a > 0, b/a is the equilibrium value.
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