QUESTION 4 Given that the function y1 = e^2x is a solution...

QUESTION 4 Given that the function $y_1 = e^{2x}$ is a solution of the corresponding homogeneous equation $(x+2)\frac{d^2y}{dx^2} - (2x+5)\frac{dy}{dx} + 2y = (x+1)e^x$, use the method of reduction to obtain the general solution of the nonhomogeneous equation.

Transcript

00:01 In this question we had given here, y double dash is plus 2y dash is equal to y plus t squared a to the power minus 2t.

00:12 So the solution to this question is, solution here is let in terms of d, in terms of d, this can be written as d square plus 2d is equal to 5 plus t square e to the power minus 2 t so the general solution is equal to general solution is equal to y c plus y p so here y c is equal to complementary function complementary function so here d square plus 2d is equal to 0 so d multiplied by d plus 2d is equal to 0 so d to 0 d1 is equal to 0 and d2 is equal to minus 2 now two roots are real and distance so complementary functions y c is equal to c1 e to the power minus 0 t plus c2 e to the power minus of 2 t so this is equate this can be written as c1 plus c2 e to the power minus 2 t this is equation number 1 now find particular integral for first part so the particular integral y p 1 is equal to 1 divided by d square plus 2d multiply by t square multiply by e to the power minus 2 t so here we will get this as e to the power minus 2 t multiply by 1 divided by d minus 2 whole square plus 2 d minus 2 multiply by t square.

02:18 So now we will get this as e to the power minus 2 t multiply by 1 divided by d square minus 4 d plus 4 minus 2d minus 4 multiplied by t squared.

02:37 So we get this as e to the power minus 2 t e to the power minus 2t multiply by 1 divided by d 2d multiply by t squared that is equal to so it becomes as e to the power minus 2 t multiply by 1 divided by d square minus 2d plus 1 minus 1 multiply by t squared.

03:09 So it is equal to, so this is equal to e to the power minus 2t multiply by 1 divided by d minus 1 1 whole square minus 1 multiply by t squared.

03:24 So here this can be written as e to the power minus 2t multiply by minus 1 multiply by 1 divided by 1 minus d minus 1 whole square multiply by t square.

03:42 So we get this as e to the power minus 2 t minus of 1 minus d minus 1 whole square whole inverse whole minus 1 multiply by t square is equal to e to the power minus 2 t minus of 1 plus d minus d minus of 1 plus d minus 1 whole square plus d minus 1 to the power 4 plus so on multiply by t square so we get here this as e to the power minus 2 t minus of 1 plus d square minus 2d plus 1 multiply by t square so we will get this as e to the power minus 2 t minus of t squared plus 2 minus 4 t plus t squared.

04:51 So this will become as e to the power minus 2 t minus of 2 t squared minus 4 t plus 2...

Question

QUESTION 4 Given that the function $y_1 = e^{2x}$ is a solution of the corresponding homogeneous equation $(x+2)\frac{d^2y}{dx^2} - (2x+5)\frac{dy}{dx} + 2y = (x+1)e^x$, use the method of reduction to obtain the general solution of the nonhomogeneous equation.

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