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Popular Calculus >

(4-t^2)y^'+2ty=3t^2

Solution

(4 - t^{2}) y^{^{'}} + 2 t y = 3 t^{2}

Solution

y = - \frac{3}{2} ln (t + 2) + \frac{3}{8} t^{2} ln (t + 2) - \frac{3}{t + 2} + \frac{3 t ^{2}}{4 ( t + 2 )} + \frac{3}{2} ln (t - 2) - \frac{3}{8} t^{2} ln (t - 2) - \frac{3}{t - 2} + \frac{3 t ^{2}}{4 ( t - 2 )} + 4 c_{1} - c_{1} t^{2}

Solution steps

(4 - t^{2}) y^{'} (t) + 2 t y = 3 t^{2}

Solve linear ODE:

y = - \frac{3}{2} ln (t + 2) + \frac{3}{8} t^{2} ln (t + 2) - \frac{3}{t + 2} + \frac{3 t ^{2}}{4 ( t + 2 )} + \frac{3}{2} ln (t - 2) - \frac{3}{8} t^{2} ln (t - 2) - \frac{3}{t - 2} + \frac{3 t ^{2}}{4 ( t - 2 )} + 4 c_{1} - c_{1} t^{2}

y = - \frac{3}{2} ln (t + 2) + \frac{3}{8} t^{2} ln (t + 2) - \frac{3}{t + 2} + \frac{3 t ^{2}}{4 ( t + 2 )} + \frac{3}{2} ln (t - 2) - \frac{3}{8} t^{2} ln (t - 2) - \frac{3}{t - 2} + \frac{3 t ^{2}}{4 ( t - 2 )} + 4 c_{1} - c_{1} t^{2}

Graph

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