Solving the above effective potential equation for the maximum under the given conditions results in a maximum potential of exactly!

Solving this differential equation, we find that the motion is described by the function

Solving this equation for r leads to an estimate of the distance

Solving the above differential equation with respect to time ( See also Binet equation ) yields:

Solving the differential equation above, a solution which is a sinusoidal function is obtained.

Solving the equation in Cartesian coordinates leads to a family of solutions known as the Hermite – Gaussian modes, while solving the equation in cylindrical coordinates leads to the Laguerre – Gaussian modes.

Solving this differential equation yields an infinite set of functions, of which the Gaussian beam is the lowest-order solution or mode.

Solving for t and replacing, the Cartesian equation would be

Solving gives, and substituting this back into the equation for yields.

Solving this equation for velocity results in the Vis-viva equation,

Solving the equation above for its unconstrained stationary points generates exactly the same stationary points as solving for the

Solving this equation for y, which is the unknown value at x, gives

Solving this equation gives roughly the following relation between the ERB value and the frequency in Hz:

Solving Poisson's equation for the potential requires knowing the charge density distribution.

Solving cubic equations is a necessary part of solving the general quartic equation, since solving the latter requires solving its resolvent cubic equation.

Solving the equation for r ( t ) is the key to the two-body problem ; general solution methods are described below.

Solving this equation for V yields the formula for exponential decay:

Solving the above equation yields

Solving the equation symbolically in MATLAB, by running

Solving the equation for and substituting with one obtains

Solving just for x < sub > 0 </ sub > requires solving a polynomial of degree 2n − 1, and solving just for γ requires solving a polynomial of degree n ( first for, then x < sub > 0 </ sub >).

* Solving in Style ( 1985 Batsford ) then ( 2002 ), Gambit Publications.

Solving, we then find that is increased slightly at the low energy scales at which the quark masses are generated by the Higgs, GeV.

Solving for the steady state then gives as the total proportion of radiation entering the second wall

Solving for the outflow velocity then gives

Solving this would involve transit sharing the rail right-of-way and splitting off the trail where Clear Creek joins Peachtree Creek, following Clear Creek around the Armour warehouse properties then tunneling under the active rail lines and I-85 to the Ansley Golf Course then rejoining the BeltLine.

In 2012, Adler contributed to a family venture when he wrote the foreword for his then 99 year old father's 87th book, " Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur on Plants ".

Solving the impedance mismatch problem for object-oriented programs starts with recognition of the differences in the specific logic systems being employed, then either the minimization or compensation of the mismatch.

Solving this, by a process known as completing the square, leads to the quadratic formula

Solving an interpolation problem leads to a problem in linear algebra where we have to solve a system of linear equations.

Solving an interpolation problem leads to a problem in linear algebra where we have to solve a matrix.

Solving the cypher leads to solving the case.

Solving the endgame often leads to a phone number, or directs Hunters to go to a certain out-of-the-way location and give a password to someone identified in a cryptic way.

Solving Apollonius ' problem iteratively in this case leads to the Apollonian gasket, which is one of the earliest fractals to be described in print, and is important in number theory via Ford circles and the Hardy – Littlewood circle method.

Solving this by building a larger model introduces a dilemma: larger models are more difficult to build and often too fragile to move smoothly.

Solving this problem mathematically, assuming that individuals are rational and have access to complete markets, Modigliani & Brumberg ( 1954 ), Albert Ando, and Milton Friedman ( 1957 ) developed what became known as the life-cycle model.

Another example would be the " Mayan Time Science " described by Carl Johan Calleman in his book Solving the Greatest Mystery of Our Time: The Mayan Calendar, which also promotes a model of unilineal evolution based on the author's interpretations of calendric cycles.

It is outlined in his book A Practical Guide for Policy Analysis: The Eightfold Path to More Effective Problem Solving, which is now in its fourth edition.

The Center for Ethical Solutions, an American bioethics think tank, is currently working on a project called " Solving the Organ Shortage ," in which it is studying the Iranian kidney procurement system in order to better inform the debate over solving the organ shortfall in the United States.

Solving crossword puzzles, which requires familiarity with a larger vocabulary, is a pastime that mature adults have long credited with keeping their minds sharp.

Solving for the free energy using the replica method, below a certain temperature, a new magnetic phase called the spin glass phase ( or glassy phase ) of the system is found to exist which is characterized by a vanishing magnetization along with a non-vanishing value of the two point correlation function between spins at the same lattice point but at two different replicas:, where are replica incides.

* Automated reasoning – Solving engines, such as used in Prolog, which produce steps to a result given a query on a fact and rule database, and automated theorem provers that aim to prove mathematical theorems with some assistance from a programmer.

* TRIZ, which is also known as Theory of Inventive Problem Solving ( TIPS ), was developed by Genrich Altshuller and his colleagues based on examining more than 200, 000 patents.

Solving Maxwell's equations for a light ray striking a boundary allows the derivation of the Fresnel equations, which can be used to predict how much of the light is reflected, and how much is refracted in a given situation.

Solving for, we see that the required stress is 3. 4 GPa, which is very close to the theoretical strength of the crystal.

He wrote an influential book " The Four Thirds Solution: Solving the Childcare Crisis in America Today ", which addresses how Shared Earning / Shared Parenting Marriage supports child development.

Solving the difficulties which arose from the Foss-Pelly slander trial took more delicate manoeuvering, but he succeeded by removing the major players in the trial from the Settlement.

Solving the version in which guards must be placed on vertices and only vertices need to be guarded is equivalent to solving the dominating set problem on the visibility graph of the polygon.

The new standards were organized around six principles ( Equity, Curriculum, Teaching, Learning, Assessment, and Technology ) and ten strands which included five content areas ( Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability ) and five processes ( Problem Solving, Reasoning and Proof, Communication, Connections, and Representation ).

Rusczyk founded and serves on the board of the nonprofit Art of Problem Solving Foundation, which he created to find ways to create opportunities for avid students of mathematics.