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

Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

(1+x)y' + y = √x

Accepted Answer

Hello. In this video, we are going to be solving for the general solution to the given linear ordinary differential equation. Now the equation given to us is the quantity 1 + X multiplied by Y + Y, and that is equal to x raised to the power of 3 halves. Now, the first thing we want to do is we want to make sure that our differential symbol Y has a coefficient of 1. So what we're going to do first is we're going to divide the entirety of this equation by 1 + X. Doing so will allow us to rewrite the differential equation as y + 1 divided by 1 + x multiplied by y equal to x raised to the power of 3 halves divided by 1 + x. Now here, the problem tells us that we are working with a linear ordinary differential equation. Now the general form of the linear differential equation is y plus a function of x p x multiplied by y equal to a function of xq x. Here px is going to be the coefficient in front of our y term, which is going to be 1 divided by 1 plus x, and q of x is the function of x that the differential equation is equal to. So, in order to simplify this differential equation, we need to first solve for integrating factor. Now the integrating factor is going to be new and that is defined as e. Raise the power of the integral of p of x. So, by identifying the PX coefficient in our differential equation, our integrating factor is going to be defined as e raised the power of the integral of 1 divided by 1 + x. Now, if we were to integrate this, this will simplify to be E raises the power of the natural logarithm of 1 + X, and by the exponential properties, this will simplify to be 1 + X. So this is going to be our integrating factor for the differential equation. Now, the integrating factor will allow us to simplify the overall differential equation. So what we're going to do is we're going to take our integrating factor, 1 + X, and multiply it to the entirety of our differential equation, which again is Y + 1 divided by 1 + XY equal to X 3 halves divided by 1 + X. So, once we distribute our integrating factor to each of the terms. Our differential equation is going to be rewritten as the quantity 1 + X multiplied by Y + Y equal to X raised to the power of 3 halves. Now, this is a little tricky to see at first, but notice that the left-hand side of the equation is in the form of a derivative. It is equivalent to the derivative of the product of 1 + x multiplied by y. So we can rewrite the left hand side of this equation as the derivative of a product of two functions. So we will have D over DX, which is a derivative with respect to X of the quantity 1 + X multiplied by Y. And that is going to equal to xrays the power of 3 halves. Now what we want to do is we want to remove the differential symbol in order to solve for Y. In order to do this, we will have to integrate both sides of this equation. So this is going to simplify the differential equation to be 1 + X multiplied by Y. And the integral of x to the 3 halves is going to be 2/5 multiplied by X raised to the 5 halves, plus the generic constant C. Now the last thing we need to do is we need to go ahead and isolate for y on the left hand side, and in order to do that, we will divide the entirety of this equation by 1 + x. So that is going to leave us with y is equal to 2 multiplied by x raised the 5 halves divided by 5 multiplied by 1 + x, plus the generic constant C divided by 1 + x. This is going to be the general solution to our differential equation. And with that being said, the overall solution to this problem is going to be option A. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

First-Order Linear EquationsSolve the differential equations in Exercises 1–14.(1+x)y' + y = √x

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

(1+x)y' + y = √x