# Undetermined Coefficients

• Apr 10th 2011, 09:48 PM
Naples
Undetermined Coefficients
Solve the initial value problem using undetermined coefficients

\$\displaystyle 2y" + 3y' - 2y = 14x^2 - 4x - 11\$ ; \$\displaystyle y(0)=0 , y'(0)=(0)\$
Use auxiliary to solve for \$\displaystyle Yc = c1e^(^1^/^2^)^x + c2e^-^2^x\$

Assume \$\displaystyle Yp = Ax^2 + Bx + C\$
\$\displaystyle Yp' = 2Ax + B\$
\$\displaystyle Yp" = 2A\$
Plug into original equation:
\$\displaystyle 4A + 6Ax + 3B - 2Ax^2 - 2Bx - 2C = 14x^2 - 4x - 11\$
\$\displaystyle -2Ax^2 = 14x^2\$
\$\displaystyle A = -7\$

\$\displaystyle 6Ax - 2Bx = -4x\$
\$\displaystyle B = -19\$

\$\displaystyle 4A + 3B - 2C = -11\$
\$\displaystyle C=-37\$

So \$\displaystyle Yp = -7x^2 - 19x - 37\$

\$\displaystyle Y = c1e^(^1^/^2^)^x + c2e^-^2^x - 7x^2 - 19x - 37\$
\$\displaystyle Y' = (1/2)c1e^(^1^/^2^)^x - 2c2e^-^2^x - 7x - 19\$
\$\displaystyle Y(0) = c1 + c2 - 37 = 0\$
\$\displaystyle c1 = 37 - c2\$
\$\displaystyle Y'(0) = (1/2)c1 - 2c2 - 19 = 0\$

\$\displaystyle (1/2)(37 - c2) - 2c2 = 19\$
\$\displaystyle c2 = -(1/5)\$
\$\displaystyle c1 = 186/5\$

\$\displaystyle Y = (186/5)e^(^1^/^2^)^x - (1/5)e^-^2^x - 7x^2 - 19x - 37\$

I'd appreciate it if someone could check through my work and make sure I didn't make an error somewhere, thanks in advance.
• Apr 10th 2011, 09:52 PM
pickslides
You can check it yourself by taking the derivative twice and substituting these results back into the original equation.

Is the original equation satisfied?