First-Order Linear Equations
Solve the differential equa...

Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

e²ˣy' + 2e²ˣ y = 2x

Accepted Answer

Welcome back everyone. Solve the differential equation eats the power of x multiplied by y plus eats the power of x multiplied by y equals 4 x. For this problem, let's notice the pattern on the left hand side we have e to the power of x multiplied by y plus. Eats the power of x can be written as eats the power of x, right, because the derivative of eats the power of x is itself multiplied by y. This is equal to fire x. And I will notice that this is a product rule of differentiation. Specifically, this is the derivative of e to the power of x multiplied by y. And we can now show that on the left hand side the derivative of e to the power of x multiplied by y is equal to 4 x. Now let's rewrite it in terms of the differential form d divided by dx off e to the power of x multiplied by y equals 4 x, and we can now multiply both sides by dx so that d of e to the power of x y equals 4 x dx. From here we can integrate both sides. On the left hand side we have the integral of e differential, which gives us e to the power of x multiplied by y itself. On the right hand side we can factor out 4 and integrate x dx using the power rule that will be x 2 divided by 2, and we're going to add a constant of integration c. 4 divided by 2 gives us 2, so we get 2 x 2 plus C. And finally, what we can do is solve the equation. Or why, right? Y equals what we can do is divide both sides by e to the power of x or basically multiply by e to the power of x. So we get e to the power of x in parentheses. 2 x 2 + c. This is our final solution. Thank you for watching.

First-Order Linear EquationsSolve the differential equations in Exercises 1–14.e²ˣy' + 2e²ˣ y = 2x

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First-Order Linear Equations
Solve the differential equations in Exercises 1–14.

e²ˣy' + 2e²ˣ y = 2x