Differential Equations - First-Order Equations - Quick Guides

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Study GuideDifferential Equations–First-Order Equations1. First-Order Homogeneous Equations1.1What Does “Homogeneous” Mean?A function(f(x, y))is calledhomogeneous of degree(n)if it satisfies this rule:for all values of(x),(y), and(z)(as long as everything is defined).What this means in simple terms:If you multiply both(x)and(y)by the same number(z), the value of the function gets multiplied by(zn).1.2Examples of Homogeneous FunctionsExample 1:Replace(x)by(zx)and(y)by(zy):So this function ishomogeneous of degree 2.Example 2:Substitute(zx)and(zy):This function ishomogeneous of degree 4.

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Study GuideExample 3:So this function ishomogeneous of degree 1.Example 4:The powers of(z)are different, so this function isnot homogeneous.Example 5:This function ishomogeneous of degree 3.1.3Homogeneous Differential EquationsA first-order differential equation of the formis calledhomogeneousif:•Both(M(x, y))and(N(x, y))are homogeneous functions•They have thesame degree

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Study GuideExample 6:•M(x, y = x2-y2(degree 2)•N(x, y) = xy) (degree 2)Since both have the same degree, the equation ishomogeneous.1.4How to Solve Homogeneous EquationsThe key idea is to use the substitution:Then,This substitution turns a homogeneous equation into aseparable equation, which is much easier tosolve.Example 7:Solving a Homogeneous EquationSolve:Step 1: Substitute(y = xv)Substitute into the equation and simplify:

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Study GuideStep 2: Separate VariablesStep 3: IntegrateStep 4: Substitute Back(v =y/x)Multiply both sides by(2x2):This is thegeneral solution.Example 8: Solving an Initial Value ProblemSolve:Step 1: Check Homogeneity•M(x, y) = 2(x + 2y)•N(x, y) = y-xBoth are degree 1 →homogeneous equation

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Study GuideStep 2: Use SubstitutionAfter simplifying, the equation becomes separable:Step 3: IntegrateUse partial fractions:Integrate:Step 4: Write the General SolutionSubstitute v = (y/x):Step 5: Apply the Initial Condition(y(1) = 0)Final Answer

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Study GuideTechnical NoteDuring separation, both sides were divided by((v + 1)(v + 2)).This means(v =-1)and(v =-2)must also be checked.Although these satisfy the differential equation, theydo not satisfy the initial condition, so they areexcluded.2. First-Order Linear Equations2.1What Is a First-Order Linear Differential Equation?A first-order differential equation is calledlinearif it can be written in the formHere:•(y)is the unknown function•(P(x))and(Q(x))are functions of(x)This form is important becauseevery equation written this way can be solved using the samemethod.2.2The Key Idea Behind the MethodThe strategy is similar to what you saw with exact equations.We multiply the entire equation by a special function called theintegrating factor, which turns theleft-hand side into the derivative of a product. Once that happens, the equation becomes easy tointegrate.Step-by-Step Solution MethodStep 1: Put the Equation in Standard FormIf needed, rewrite the equation so it looks like:

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Study GuideStep 2: Find the Integrating FactorThe integrating factor is defined asStep 3: Multiply the Entire Equation by(μ(x))After multiplying, the equation becomes:The left-hand sidealways collapsesinto a single derivative:Step 4: Integrate Both SidesIntegrating gives:Important ReminderDonottry to memorize the final formula.Instead, rememberwhythe method works and follow the steps carefully.Example 1:A Basic Linear EquationSolve:

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Study GuideStep 1: Identify(P(x))and(Q(x))•P(x) = 2•Q(x) = xStep 2: Find the Integrating FactorStep 3: Multiply the EquationThe left side becomes:Step 4: IntegrateExample 2:An Initial Value Problem (IVP)Solve:Step 1: Find the Integrating Factor

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Study GuideStep 2: Multiply the EquationThis becomes:Step 3: IntegrateStep 4: Apply the Initial ConditionFinal AnswerExample 3:A Linear Equation That Needs RewritingSolve:

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Study GuideStep 1: Rewrite in Standard FormStep 2: Find the Integrating FactorStep 3: Multiply and IntegrateFinal AnswerExample 4:Solving Two Similar Equations(a)(b)Both equations have:

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