Math 340 – Differential Equations (I)
Review for Exam 1
Exam 1 will consist in 5 problems, similar to homework problems.
Review the theory for each section, and the homework problems.
1.1 Differential equations and mathematical models 13, 32
- nth order DE; initial condition; IVP;
- Particular solution; general solutions;
- Writing a mathematical model for a given system.
1.2 Integrals as general and particular solutions: 1, 11
- General and particular solutions; integral forms for solutions;
- Applications: position function, velocity, acceleration;
1.3 Direction fields: 11, 21
- Slope fields, solution curves;
- Solutions are tangent to direction fields; constructing solution curves graphically;
- Theorem of existence and uniqueness of solutions.
1.4 Separable equations: 19
- Definition and solving; implicit & explicit solutions; singular solutions;
1.5 Linear 1st order DE: 3, 30
- Normal form, integrating factor & solving; existence & uniqueness theorem.
- Mixture problems.
1.6 Substitutions methods: 1,3,9, 21,31,43
- Substitution y=F(x,v), change of variable v=G(x,y)
- Homogeneous DE, v=y/x => separable eq.
- Bernoulli, v=y1-n => linear eq.
- Exact equations:
- Standard form, differential form; definition (condition);
- Method of solving exact DE.
- Reduction of 2nd ODE F(x,y,y’,y’’)=0
- Missing x: substitution p=y’;
- Missing y: subst. p=y’ => y’’=p dp/dx
1.7 Population models (Modeling -> DE & solving): 1,3,6,11
- General population equation, birth/death rates;
- Natural growth;
- Logistic growth (separable);
- Finding the analytic solution;
- Equilibrium solutions & carrying capacity;
1.8 Acceleration-velocity models (Modeling -> DE & solving): 1,3
- Constant gravity (integrate);
- Resistance proportional to v or v2 (separable);
Math 340 – Differential Equations (I)
Review for Exam 2
Exam 2 will consist in 5 problems. Review the theory for each section and the homework problems.
2.1+2.2 General solution of linear equations: 2.1/27; 2.2/7, 21
- Homogeneous and non-homogeneous linear DE; linear operator;
- Principle of superposition; existence and uniqueness of solutions of IVP;
- Linear dependence of functions, Wronskian;
- General solution: yc=c1y1+…+cn yn, Y=yc+yp.
2.3 Homogeneous equations with constant coefficients: 21, 39;
- Polynomial operators, characteristic equation, distinct & repeated roots;
- Complex-valued functions, Euler’s formula, Re & Im parts; complex roots and solutions;
2.4+2.6 Mechanical vibrations: 2.4/3, 15; 2.6/15
- Terminology: mass-spring system, amplitude, circular frequency, phase angle, etc.
- Forced oscillations and resonance: natural frequency; transient solution, steady state solution;
2.5 Undetermined coefficients and variation of parameters: 21, 47
- Method of undetermined coefficients: f(x)=sum of P(x)erx (r complex);
- No duplication strategy for yp;
- Duplication: definition & strategy for picking a particular solution;
- Variation of parameters: the idea (yc=cy => yp=uy) and using the formula.
2.7 Electrical circuits: 1, 7
- Terminology: resistor, inductor, capacitor, voltage drop; LQ’’+RQ’+(1/C)Q=E(t)
- Mechanical-electrical analogy; transient and steady periodic current; electric resonance.
2.8 Endpoint problems and eigenvalues: 1, 6
- Boundary value problem, eigenvalue problem; eigenvalues, eigenfunctions.
3.1 Power series 1, 11
- Power series: definition, interval and radius of convergence (Theorem); ratio test; examples: exp, sin, cos;
- Power series representations of functions, Taylor series; identity principle
- Operations with p.s.: +,-,x,/; differentiation and integration (Theorem: R does not decrease);
- Power Series Method: deriving the recurrence relation, computing ROC;
- Recognizing power series representations of known functions;
3.2 Series solutions near ordinary points y’’+P(x)y’+Q(x)y=0 : 1, 17
- Ordinary/singular points; Theorem: 2 independent solutions & ROC=distance to singular points
3.3 Regular singular points: 1, 9, 17
- Types of singular points: regular or not regular
- Method of Frobenius
- Equidimensional equation, indicial equation and exponents (its roots)
- Frobenius series solutions: r1-r2 not in Z => 2 indep. sol. y=xr(c0+c1x+…)
- Finding Frobenius series.
Math 340 – Differential Equations (I)
Review for Exam 3
Exam 3 will consist in 5 problems. Review the theory for each section, and the homework problems.
4.1 Laplace Transform and Inverse Transform: 2, 11, 25

- Laplace transform:L(f)(s)=  e  st f (t )dt , properties: linear & 1:1 (f,g cont.) => invertible on its image
0
- Inverse Laplace Transform: definition, properties: linear & 1:1 (by definition this time).
- Classes of functions: piece-wise continuous, piece-wise smooth, exponential order.
- “Existence of LT” Theorem (f pws & exp.); “Uniqueness of LT” Theorem (existence of Inverse LT)
4.2 Transformation of IVP: 1, 11,19,22
- Transform of Derivatives Theorem L[f’]=sL[f]-f(0); transforms of higher derivatives: L(f(n))=…
- Solving IVP using Laplace Transform (4.2/1)
- Transforms of Integrals: L   = M1/s  L
- Finding New LP from Old (find a DE satisfied by L(f) …) (4.2/11, 19)
4.3 Translation and Partial Fractions: 1, 5, 11
- Partial fractions decomposition
- Translation on the s-Axis Theorem: L M(eat) =T-a L
4.4 Derivatives, Integrals and Product of Transforms: 1, 15, 23, 29
t
- Convolution: (f*g)(t)=  f ( x) g (t  x)dx ; properties (linear, commut.); Convolution Theorem (L * =  L)
0
- Differentiation of transforms Theorem (d/ds L=L M-t);
- Integration of transforms Theorem ( L= L M1/t)
4.5 Periodic and piecewise continuous forcing functions
- Unit step functions and their Laplace transforms, translation on t-axis;
- Laplace transform of a periodic function
5.1: Linear systems of DE
- Reducing to 1st order systems: 5.1/1; homogeneous & non-homogeneous linear systems;
- Existence and uniqueness of solutions of x’=P(t)x+f(t)
5.2 The method of elimination:
- Linear differential operators, operator determinant;
- Elimination method: System -> Equation 5.2/1
5.3 Linear systems and matrices
- Matrices and operations: +, x, d/dt;
- Linear systems: 5.3/21
- Homogeneous: principle of superposition, linear independence, Wronskian;
- General solution of homogeneous systems: x(t)=X(t) c (lin. combin. of n-indep. sol.)
- Solution of nonhomogeneous systems: x(t)=xc(t)+xp(t) (complementary function); 5.3/31
5.4 The eigenvalue method for homogeneous systems X’=AX : 5.4/1
- Eigenvalues, eigenvector, characteristic equation; the eigenvalue method.
5.7 Matrix exponentials and linear homogeneous systems 5.7/1,15
- Exponential matrices eA; If AB=BA => eA+B=eA eB;
- Matrix exponential solutions: x(t)=eAtx0; computing eAt:
5.8 Nonhomogeneous linear systems 5.8/1, 21
- Undetermined coefficients (xp(t)=lin.combination of f, f’,…); eliminating duplication w. xc(t)
- Fundamental matrix, fundamental matrix solutions x(t)=(t)c, x(t)= (t) -1(a)b
- Variation of parameters: xp(t)= (t)u(t), u(t)= -1(t)f(t)dt; IVP: x=eAt(x0+e-tAf(t)dt)
Math 340 – Differential Equations (I)
Review for Final Exam
The Final Exam will be comprehensive, consisting in 7 to 10 problems.
You are allowed one page (front and back) with formulas, but no solutions to specific problems.
Please review the theory first (see Review Guides for Tests 1,2,3 & final), starting with:
1) “Title concepts” (example: 1.3 Direction fields etc.):
- Definition, criterion (if any) and properties:
- Review them (formal def., a graphical representation and a good example)
- Check (closed book/notes) that you’ve got it!
2) Review (even better: rework closed book/notes) associated homework problems:
- Computational problems
- Easy proofs of simple corollaries/properties (if any).
3) For an A in the final:
- Review proofs for the theorems listed (if any)
- Rework (without book/notes or HWk) all the other homework problems.
Make sure you know how to input matrices and finding eigenvalues eigVl(mat) and eigenvectors
eigVc(mat).
In addition to the previous review sheets:
Ch. 6
- Euler method for DE and systems: 6.1/1
- Improved Euler method for DE and systems: 6.2/1
- Terminology: Current Point, Predicted Point, Improved Euler Point;
- Formulas: predictor, corrector.
- Runge-Katta method (idea):
- Simpson’s formula for numerical integration
- Euler’s method for estimating slopes
- Errors:
- Absolute and relative error: definition and computing it;
- Numerical methods of order k;
- Understanding the content of the theorems:
- Euler’s method is of order 1;
- Improved Euler’s method is of order 2.
- RK-method is of order 4
- Numerical method for systems
- Vector valued analogs: Euler’s method and RK-method
- Variable step size methods