2 Homogeneous second order ODE's with arbitrary coefficient:...

2 Homogeneous second order ODE's with arbitrary coefficient: (40%) Find the complete solutions of the homogeneous equations in form of Euler-Cauchy Equation or reduction of order: (a) $x^2y'' + 4xy' + 2y = 0$; $x > 0$, $y(1) = 0$, $y'(1) = 1$ (10%) (b) $x^2y'' + 2xy' + \frac{1}{4}y = 0$; $x > 0$, $y(1) = 1$, $y'(1) = \frac{3}{2}$ (10%) (c) $x^2y'' - 3xy' + 5y = 0$; $x > 0$, $y(1) = 2$, $y(e^{\frac{\pi}{2}}) = e^{\frac{\pi}{2}}$ (10%) (d) Find general solution for: $x(x-1)y'' - xy' + y = 0$; (hint: find a simple $y_1(x)$ by observation, then try order reduction) (10%)

Transcript

00:01 Hey guys in this question we have been asked to find homogeneous solutions so first question is given to us which is y double so y double as plus 4 y dash so this is 4 y dash plus 4 y equals to x squared sine h 2 x so we can write h sign h 2 x as x squared as x squared multiplied with e power 2x minus e power minus 2 x divided y 2 so this is e power minus 2 x e power minus 2 x and we can write it as x square e power 2x minus 1 divided 2 x square e power minus 2x and this is also divided by 2 so y double dash plus 4 y dash plus 4 y this is equal to so the auxiliary equation will be m square plus 4m plus 4 equals to 0 so this is auxiliary equation so this is our auxiliary equation so this is going to be m square plus 4m plus 4m equals to this is m plus 2 poly square this is equal to so here m is equal to so this is m so m equals to minus 2 and minus 2 so m has 2 values which is minus 2 and minus 2 so therefore the complementary equation will be complementary equation complementary equation silly y c equals to c1 plus c2 x so this is let me just read that c1 plus c2 c2 c1 plus c2 e power minus 2 x where where c1 and c2 are auxiliary auxiliary constants so this is our complementary equation and so the particular equation will be so let me write that so the particular equation will be y p equals to a 1 x square plus a 2x plus a 3 multiplied with e power 2x plus b1 x squared plus b2x multiplied with x square e power minus 2 x where a 1 a 2 a 3 a 1 a 2 so this is a 1 a 2 a 3 and b1 b2 and b3 are constant and we have to determine it and in this question we don't have to determine it this is our final answer for first part and the second part for second part we have been given the equation as y double dash 5x plus 2y dash plus 5i this is our differential equation plus 5i equals to 2 sine 2x plus 2 e power minus x cos so this is cost 2 so for this auxiliary equation will be so auxiliary equation will be m square plus 2m m square plus 2m plus 5 equals to 0 so here m equals to going to be m equals to minus 2 plus minus under root 4 minus 20 you are to die 2 and this is going to be minus 2 plus minus under root minus 16 divided by 2 which is a complex number so we can write it as m is going to be minus 1 plus minus 2 iota and here so m has 2 values one is m equals to minus 1 plus 2 iota and the second one is 1 minus 2 aorta so these are two values of m for from auxiliary equations so the complementary solution will be so the complementary complementary solution will be yc equals to c1 cost 2x plus c2 sine.

04:47 So let me see 2 sine 2x multiplied with e power minus x where c1 and c2 are c1 and c2 are auxiliary constant.

05:05 So this is going to be so so after that.

05:08 The particular solution will be so particular solution particular solution will be y p is going to be a1 shine 2 x so this is a 1 sine 2 x plus a 2 x plus a 2 x plus e power minus x multiplied with b1 shine 2 x plus b2 cos 2 x so this is our particular solution when where a1 and we know that a1, a2, b1, b2 are constant...

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