42–43. Implicit solutions for separable equations For the foll...

Question

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

a. Find the general solution of the equation.

e⁻ʸᐟ²y'(x) = 4x sin x² − x; y(0) = 0, y(0) = ln(1/4), y(√(π/2)) = 0

Accepted Answer

Welcome back, everyone. Find the general solution to the separable equation. It's the power of negative W divided by 4, multiplied by W S equals 8S of Squad minus 2S. For this problem we're going to rewrite the W of S as a function Y of S for simplicity. Therefore, if we consider our original equation, we can state that it has a form of E raise the power of negative or Y divided by 4. Multiplied by the derivative. D Y divided by D S. Equals 8s. Sign of squad minus 2s. And this is the differential equation that we want to solve. So now what we're going to do is simply separate the variables. Let's go ahead and multiply both sides by DS. We're going to get E to the power of negative Y divided by 4 is equal to 8S multiplied by sign of Squad minus 2s, the right hand side is multiplied by DS, and the left hand side is multiplied by D Y, right? So now we have successfully separated the variables, and we can integrate both sides. So we're going to add the integral sign on the left-hand side and the right hand side. Let's go ahead and integrate the function on the left. The integral of E erase the power of X is E is Ease the power of X. So, in this context, we're going to get Ease the power of negative Y divided by 4. But because we have a composite function. That is. A linear function within our exponent, according to the reverse chain rule, we are also dividing by the slope of the linear term, and our slope is -1/4. So on the left hand side, our integral is Ease the power of negative Y divided by 4, divided by -14, right? Considering the right hand side, we're going to split our integral into two parts, and first of all, let's go ahead and integrate 8S multiplied by a sign of Squared. What we can do is simply introduce u substitution. Let's suppose that u equals squared. Then the derivative of u with respect to S is 2 S. And that we can show that as Multiplied by DS is equal to DU divided by 2 or basically 12 DU. So we can rewrite our integral as, first of all, let's factor out 8 because it's a constant. S multiplied by DS is 1/2 the U, and then we get sin of U, right? So sine of U is multiplied by 1/2 DU. Now 8 divided by 2 is 4, we can combine our constants. And the integral of sine is negative cosine, so we get -4, cosine of U plus a constant of integration C. So now for the right hand side, we can already say that our integral consists of negative for, cosine of. Now U is Squad, so we get -4 cosine of Squad, and then we subtract 2 multiplied by the integral of S. The integral of S is S2 divided by 2 according to the power rule. And then we're going to add a constant of integration C for an indefinite integral. From here, we're going to simplify the expression. And we're going to find the general solution. So, let's go ahead and simplify everything, starting with the left hand side. On the left hand side, we can rewrite it as -4 E to the power of negative Y divided by 4. On the right hand side, we have -4, cosine of S2 minus S2 + C. What we can do from here is multiply both sides by -1. So we get 4 E to the power of negative Y divided by 4 equals 4 cosine of Squad. Plus squad minus C. So we can label our previous constant as C1. And the new one is C2, right? We can just say that we're adding C2, instead of subtracting C1, we can just say we're adding a new constant C2 such that C2 is equal to negative C1. Now, let's go ahead and divide both sides by 4. If we divide both sides by 4, we will be able to get the expression for E to the power of negative Y divided by 4. So, on the left-hand side, we have each, the power of negative Y divided by 4. On the right hand side, we're going to end up with cosine of S2 + S2 divided by 4 plus C2 divided by 4. So now we can show that C2 divided by 4 is just a new constant such as C3, right? Eats the power of negative y divided by 4 is equal to. Cosine of S2 plus S2 divided by 4 + C3, where C3 is equal to C2 divided by 4. Well then, what we can do now is basically take natural log of both sides to solve for negative Y divided by 4. So negative Y divided by 4 is equal to LN of the right hand side. So LN of cosine of S2 plus. As square divided by 4 plus. C 3 And from here we can solve for Y, right? We can essentially multiply both sides by -4. So now we can show that Y equals -4 LN of cosine of S2 + 2 divided by 4 + C3. And then we can just state that C3 is simply equal to C, right, not to have any subscript. Well then we have our final answer. Let's label it and thank you for watching.

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.
a. Find the general solution of the equation.

e⁻ʸᐟ²y'(x) = 4x sin x² − x; y(0) = 0, y(0) = ln(1/4), y(√(π/2)) = 0

Key Concepts

Separable Differential Equations

Implicit Solutions

Initial Conditions and Particular Solutions

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