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Differential Equations - First-Order Equations

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    Study Guide Differential Equations – First - Order Equations 1. First - Order Homogeneous Equations 1.1 What Does “Homogeneous” Mean? A function ( f(x, y) ) is called homogeneous 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 ( z n ) . 1.2 Examples of Homogeneous Functions Example 1 : Replace ( x ) by ( zx ) and ( y ) by ( zy ) : So this function is homogeneous of degree 2 . Example 2 : Substitute ( zx ) and ( zy ) : This function is homogeneous of degree 4 .

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    Study Guide Example 3 : So this function is homogeneous of degree 1 . Example 4 : The powers of ( z ) are different, so this function is not homogeneous . Example 5 : This function is homogeneous of degree 3 . 1.3 Homogeneous Differential Equations A first - order differential equation of the form is called homogeneous if: • Both ( M(x, y) ) and ( N(x, y) ) are homogeneous functions • They have the same degree

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    Study Guide Example 6 : • M(x, y = x 2 - y 2 (degree 2) • N(x, y) = xy) (degree 2) Since both have the same degree, the equation is homogeneous . 1.4 How to Solve Homogeneous Equations The key idea is to use the substitution: Then, This substitution turns a homogeneous equation into a separable equation , which is much easier to solve. Example 7: Solving a Homogeneous Equation Solve: Step 1: Substitute ( y = xv ) Substitute into the equation and simplify:

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    Study Guide Step 2: Separate Variables Step 3: Integrate Step 4: Substitute Back ( v = y/x ) Multiply both sides by ( 2x 2 ) : This is the general solution . Example 8: Solving an Initial Value Problem Solve: Step 1: Check Homogeneity • M(x, y) = 2(x + 2y) • N(x, y) = y - x Both are degree 1 → homogeneous equation

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    Study Guide Step 2: Use Substitution After simplifying, the equation becomes separable: Step 3: Integrate Use partial fractions: Integrate: Step 4: Write the General Solution Substitute v = (y/x): Step 5: Apply the Initial Condition ( y(1) = 0 ) Final Answer

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    Study Guide Technical Note During 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, they do not satisfy the initial condition , so they are excluded. 2. First - Order Linear Equations 2.1 What Is a First - Order Linear Differential Equation? A first - order differential equation is called linear if it can be written in the form Here: • ( y ) is the unknown function • ( P(x) ) and ( Q(x) ) are functions of ( x ) This form is important because every equation written this way can be solved using the same method . 2.2 The Key Idea Behind the Method The strategy is similar to what you saw with exact equations. We multiply the entire equation by a special function called the integrating factor , which turns the left - hand side into the derivative of a product. Once that happens, the equation becomes easy to integrate. Step - by - Step Solution Method Step 1: Put the Equation in Standard Form If needed, rewrite the equation so it looks like:

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    Study Guide Step 2: Find the Integrating Factor The integrating factor is defined as Step 3: Multiply the Entire Equation by ( μ (x) ) After multiplying, the equation becomes: The left - hand side always collapses into a single derivative: Step 4: Integrate Both Sides Integrating gives: Important Reminder Do not try to memorize the final formula. Instead, remember why the method works and follow the steps carefully. Example 1: A Basic Linear Equation Solve:

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    Study Guide Step 1: Identify ( P(x) ) and ( Q(x) ) • P(x) = 2 • Q(x) = x Step 2: Find the Integrating Factor Step 3: Multiply the Equation The left side becomes: Step 4: Integrate Example 2: An Initial Value Problem (IVP) Solve: Step 1: Find the Integrating Factor

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    Study Guide Step 2: Multiply the Equation This becomes: Step 3: Integrate Step 4: Apply the Initial Condition Final Answer Example 3: A Linear Equation That Needs Rewriting Solve:

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    Study Guide Step 1: Rewrite in Standard Form Step 2: Find the Integrating Factor Step 3: Multiply and Integrate Final Answer Example 4: Solving Two Similar Equations (a) (b) Both equations have:

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